{"id":4255,"date":"2018-12-11T14:03:09","date_gmt":"2018-12-11T19:03:09","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/finding-composite-and-inverse-functions\/"},"modified":"2019-01-17T14:15:36","modified_gmt":"2019-01-17T19:15:36","slug":"finding-composite-and-inverse-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/finding-composite-and-inverse-functions\/","title":{"raw":"Finding Composite and Inverse Functions","rendered":"Finding Composite and Inverse Functions"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\">\r\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Find and evaluate composite functions<\/li>\r\n \t<li>Determine whether a function is one-to-one<\/li>\r\n \t<li>Find the inverse of a function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836288458\" class=\"be-prepared\">\r\n<p id=\"fs-id1167836595563\">Before you get started, take this readiness quiz.<\/p>\r\n\r\n<ol id=\"fs-id1167836440251\" type=\"1\">\r\n \t<li>If \\(f\\left(x\\right)=2x-3\\) and \\(g\\left(x\\right)={x}^{2}+2x-3,\\) find \\(f\\left(4\\right).\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/5e548626-8f0f-496d-ab87-4f0358ca2fd3#fs-id1167836521479\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Solve for \\(x,\\) \\(3x+2y=12.\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/b03538a1-8a7b-4158-a68b-e0e8a24c9fd4#fs-id1167835229496\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Simplify: \\(5\\frac{\\left(x+4\\right)}{5}-4.\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836390100\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1167829579375\">In this chapter, we will introduce two new types of functions, exponential functions and logarithmic functions. These functions are used extensively in business and the sciences as we will see.<\/p>\r\n\r\n<div class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Find and Evaluate Composite Functions<\/h3>\r\n<p id=\"fs-id1167829747783\">Before we introduce the functions, we need to look at another operation on functions called <span data-type=\"term\" class=\"no-emphasis\">composition<\/span>. In composition, the output of one function is the input of a second function. For functions \\(f\\) and \\(g,\\) the composition is written \\(f\\circ g\\) and is defined by \\(\\left(f\\circ g\\right)\\left(x\\right)=f\\left(g\\left(x\\right)\\right).\\)<\/p>\r\nWe read \\(f\\left(g\\left(x\\right)\\right)\\) as \\(\\text{\u201c}f\\) of \\(g\\) of \\(x\\text{.\u201d}\\)\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836294921\" data-alt=\"This figure shows x as the input to a box denoted as function g with g of x as the output of the box. Then, g of x is the input to a box denoted as function f with f of g of x as the output of the box.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows x as the input to a box denoted as function g with g of x as the output of the box. Then, g of x is the input to a box denoted as function f with f of g of x as the output of the box.\" \/><\/span>\r\n<p id=\"fs-id1167833212955\">To do a composition, the output of the first function, \\(g\\left(x\\right),\\) becomes the input of the second function, <em data-effect=\"italics\">f<\/em>, and so we must be sure that it is part of the domain of <em data-effect=\"italics\">f<\/em>.<\/p>\r\n\r\n<div data-type=\"note\">\r\n<div data-type=\"title\">Composition of Functions<\/div>\r\n<p id=\"fs-id1167836417281\">The composition of functions <em data-effect=\"italics\">f<\/em> and <em data-effect=\"italics\">g<\/em> is written \\(f\u00b7g\\) and is defined by<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167829930414\" class=\"unnumbered\" data-label=\"\">\\(\\left(f\\circ g\\right)\\left(x\\right)=f\\left(g\\left(x\\right)\\right)\\)<\/div>\r\n<p id=\"fs-id1167829791406\">We read \\(f\\left(g\\left(x\\right)\\right)\\) as \\(f\\) of \\(g\\) of <em data-effect=\"italics\">x<\/em>.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167836628588\">We have actually used composition without using the notation many times before. When we graphed quadratic functions using translations, we were composing functions. For example, if we first graphed \\(g\\left(x\\right)={x}^{2}\\) as a parabola and then shifted it down vertically four units, we were using the composition defined by \\(\\left(f\\circ g\\right)\\left(x\\right)=f\\left(g\\left(x\\right)\\right)\\) where \\(f\\left(x\\right)=x-4.\\)<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167826024537\" data-alt=\"This figure shows x as the input to a box denoted as g of x equals x squared with x squared as the output of the box. Then, x squared is the input to a box denoted as f of x equals x minus 4 with f of g of x equals x squared minus 4 as the output of the box.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows x as the input to a box denoted as g of x equals x squared with x squared as the output of the box. Then, x squared is the input to a box denoted as f of x equals x minus 4 with f of g of x equals x squared minus 4 as the output of the box.\" \/><\/span>\r\n<p id=\"fs-id1167829851372\">The next example will demonstrate that \\(\\left(f\\circ g\\right)\\left(x\\right),\\) \\(\\left(g\\circ f\\right)\\left(x\\right)\\) and \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) usually result in different outputs.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167829718921\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836628343\">\r\n<div data-type=\"problem\" id=\"fs-id1167824781200\">\r\n<p id=\"fs-id1167836409476\">For functions \\(f\\left(x\\right)=4x-5\\) and \\(g\\left(x\\right)=2x+3,\\) find: <span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(x\\right),\\) <span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(x\\right),\\) and <span class=\"token\">\u24d2<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836685094\">\r\n<p id=\"fs-id1167829860686\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167836449256\" class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of f of g of x to obtain that f of g of x equals f of g of x. In the second step, we substitute 2 x plus 3 for g of x. This means that f of g of x equals f of 2 x plus 3. In the third step, we find f of 2 x plus 3 where f of x equals 4 x minus 5. This means that f of g of x equals 4 times the quantity 2 x plus 3, minus 5. In the fourth step, we distribute. This means that f of g of x equals 8 x plus 12 minus 5. In the fifth step, we simplify. This means that f of g of x equals 8 x plus 7.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Use the definition of \\(\\left(f\\circ g\\right)\\left(x\\right).\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829850078\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836507880\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836714410\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829787748\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167836424078\" class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of g of f of x to obtain that g of f of x equals g of f of x. In the second step, we substitute 4 x minus 5 for f of x. This means that g of f of x equals g of 4 x minus 5. In the third step, we find g of 4 x minus 5 where g of x equals 2 x plus 3. This means that g of f of x equals 2 times the quantity 4 x minus 5, plus 3. In the fourth step, we distribute. This means that g of f of x equals 8 x minus 10 plus 3. In the fifth step, we simplify. This means that g of f of x equals 8 x minus 7.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Use the definition of \\(\\left(f\\circ g\\right)\\left(x\\right).\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832999727\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829907616\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824763273\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836485988\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836417376\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829620750\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832940195\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167833136344\">Notice the difference in the result in part <span class=\"token\">\u24d0<\/span> and part <span class=\"token\">\u24d1<\/span>.<\/p>\r\n<span class=\"token\">\u24d2<\/span> Notice that \\(\\left(f\u00b7g\\right)\\left(x\\right)\\) is different than \\(\\left(f\\circ g\\right)\\left(x\\right).\\) In part <span class=\"token\">\u24d0<\/span> we did the composition of the functions. Now in part <span class=\"token\">\u24d2<\/span> we are not composing them, we are multiplying them.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cccccc}\\text{Use the definition of}\\phantom{\\rule{0.2em}{0ex}}\\left(f\u00b7g\\right)\\left(x\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\left(f\u00b7g\\right)\\left(x\\right)=f\\left(x\\right)\u00b7g\\left(x\\right)\\hfill \\\\ \\text{Substitute}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right)=4x-5\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right)=2x+3.\\hfill &amp; &amp; &amp; &amp; &amp; \\left(f\u00b7g\\right)\\left(x\\right)=\\left(4x-5\\right)\u00b7\\left(2x+3\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; &amp; &amp; \\left(f\u00b7g\\right)\\left(x\\right)=8{x}^{2}+2x-15\\hfill \\end{array}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836613359\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836576079\">\r\n<div data-type=\"problem\" id=\"fs-id1167829650480\">\r\n\r\nFor functions \\(f\\left(x\\right)=3x-2\\) and \\(g\\left(x\\right)=5x+1,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(x\\right)\\) <span class=\"token\">\u24d2<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right)\\).\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836699034\">\r\n<p id=\"fs-id1167833047859\"><span class=\"token\">\u24d0<\/span>\\(15x+1\\)<span class=\"token\">\u24d1<\/span>\\(15x-9\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(15{x}^{2}-7x-2\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167824590525\" class=\"try\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167824735112\">\r\n<p id=\"fs-id1167836299504\">For functions \\(f\\left(x\\right)=4x-3,\\) and \\(g\\left(x\\right)=6x-5,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(x\\right),\\) <span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(x\\right),\\) and <span class=\"token\">\u24d2<\/span> \\(\\left(f\u00b7g\\right)\\left(x\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836447318\">\r\n<p id=\"fs-id1167836319241\"><span class=\"token\">\u24d0<\/span>\\(24x-23\\)<span class=\"token\">\u24d1<\/span>\\(24x-23\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(24{x}^{2}-38x+15\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the next example we will evaluate a composition for a specific value.\r\n<div data-type=\"example\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167833086943\">\r\n\r\nFor functions \\(f\\left(x\\right)={x}^{2}-4,\\) and \\(g\\left(x\\right)=3x+2,\\) find: <span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(-3\\right),\\) <span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(-1\\right),\\) and <span class=\"token\">\u24d2<\/span> \\(\\left(f\\circ f\\right)\\left(2\\right).\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829580345\">\r\n<p id=\"fs-id1167836521713\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167829628226\" class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of f of g of negative 3 to obtain that f of g of negative 3 equals f evaluated at g of negative 3. In the second step, we find g of negative 3 where g of x equals 3 x plus 2. This means that f of g of negative 3 equals f evaluated at 3 times negative 3 plus 2. In the third step, we simplify to obtain that f of g of negative 3 equals f of negative 7. In the fourth step, we find f of negative 7 where f of x equals x squared minus 4. This means that f of g of negative 3 equals negative 7 squared minus 4. In the fifth step, we simplify to obtain that f of g of negative 3 equals 45.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Use the definition of \\(\\left(f\\circ g\\right)\\left(-3\\right).\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836698636\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833009341\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829807742\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<table class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of g of f of negative 1 to obtain that g of f of negative 1 equals g evaluated at f of negative 1. In the second step, we find f of negative 1 where f of x equals x squared minus 4. This means that g of f of negative 1 equals g of negative 1 squared minus 4. In the third step, we simplify. This means that g of f of negative 1 equals g of negative 3. In the fourth step, we find g of negative 3 where g of x equals 3 x plus 2. This means that g of f of negative 1 equals 3 times negative 3 plus 2. In the fifth step, we simplify. This means that g of f of negative 1 equals negative 7.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Use the definition of \\(\\left(g\\circ f\\right)\\left(-1\\right).\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836623006\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829589897\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830123199\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167829634197\" class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of f of f of 2 to obtain that f of f of 2 equals f evaluated at f of 2. In the second step, we find f of x where f of x equals x squared minus 4. This means that f of f of 2 equals f of 2 squared minus 4. In the third step, we simplify to obtain that f of f of 2 equals f of 0. In the fourth step, we find f of 0 where f of x equals x squared minus 4. This means that f of f of 2 equals 0 squared minus 4. In the fifth step, we simplify to obtain that f of f of 2 equals negative 4.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Use the definition of \\(\\left(f\\circ f\\right)\\left(2\\right).\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836319358\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836730743\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836325800\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836620982\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829579755\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836418899\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833087107\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836319449\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829744245\">\r\n<div data-type=\"problem\" id=\"fs-id1167829908684\">\r\n\r\nFor functions \\(f\\left(x\\right)={x}^{2}-9,\\) and \\(g\\left(x\\right)=2x+5,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(-2\\right),\\) <span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(-3\\right),\\) and <span class=\"token\">\u24d2<\/span> \\(\\left(f\\circ f\\right)\\left(4\\right).\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n\r\n<span class=\"token\">\u24d0<\/span> \u20138 <span class=\"token\">\u24d1<\/span> 5 <span class=\"token\">\u24d2<\/span> 40\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836561320\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167833051710\">\r\n<div data-type=\"problem\" id=\"fs-id1167829579215\">\r\n<p id=\"fs-id1167829692973\">For functions \\(f\\left(x\\right)={x}^{2}+1,\\) and \\(g\\left(x\\right)=3x-5,\\) find <span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(-1\\right),\\) <span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(2\\right),\\) and <span class=\"token\">\u24d2<\/span> \\(\\left(f\\circ f\\right)\\left(-1\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836688612\">\r\n<p id=\"fs-id1167829595371\"><span class=\"token\">\u24d0<\/span> 65 <span class=\"token\">\u24d1<\/span> 10 <span class=\"token\">\u24d2<\/span> 5<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Determine Whether a Function is One-to-One<\/h3>\r\n<p id=\"fs-id1167830093543\">When we first introduced functions, we said a <span data-type=\"term\" class=\"no-emphasis\">function<\/span> is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each <em data-effect=\"italics\">x<\/em>-value is matched with only one <em data-effect=\"italics\">y<\/em>-value.<\/p>\r\n<p id=\"fs-id1167836340795\">We used the birthday example to help us understand the definition. Every person has a birthday, but no one has two birthdays and it is okay for two people to share a birthday. Since each person has exactly one birthday, that relation is a function.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167829586481\" data-alt=\"This figure shows two tables. To the left is the table labeled Name, which from top to bottom reads Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, and Danny. The table on the right is labeled Birthday, which from top to bottom reads January 12, February 3, April 25, May 10, May 23, July 24, August 2, and September 15. There are arrows going from Alison to April 25, Penelope to May 23, June to August 2, Gregory to September 15, Geoffrey to January 12, Lauren to May 10, Stephen to July 24, Alice to February 3, Liz to July 24, and Danny to no birthday.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_008_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two tables. To the left is the table labeled Name, which from top to bottom reads Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, and Danny. The table on the right is labeled Birthday, which from top to bottom reads January 12, February 3, April 25, May 10, May 23, July 24, August 2, and September 15. There are arrows going from Alison to April 25, Penelope to May 23, June to August 2, Gregory to September 15, Geoffrey to January 12, Lauren to May 10, Stephen to July 24, Alice to February 3, Liz to July 24, and Danny to no birthday.\" \/><\/span>\r\n<p id=\"fs-id1167829789822\">A function is <span data-type=\"term\">one-to-one<\/span> if each value in the range has exactly one element in the domain. For each ordered pair in the function, each <em data-effect=\"italics\">y<\/em>-value is matched with only one <em data-effect=\"italics\">x<\/em>-value.<\/p>\r\n<p id=\"fs-id1167836516310\">Our example of the birthday relation is not a one-to-one function. Two people can share the same birthday. The range value August 2 is the birthday of Liz and June, and so one range value has two domain values. Therefore, the function is not one-to-one.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836543477\">\r\n<div data-type=\"title\">One-to-One Function<\/div>\r\nA function is <strong data-effect=\"bold\">one-to-one<\/strong> if each value in the range corresponds to one element in the domain. For each ordered pair in the function, each <em data-effect=\"italics\">y<\/em>-value is matched with only one <em data-effect=\"italics\">x<\/em>-value. There are no repeated <em data-effect=\"italics\">y<\/em>-values.\r\n\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167836548952\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167829589807\">\r\n\r\nFor each set of ordered pairs, determine if it represents a function and, if so, if the function is one-to-one.\r\n\r\n<span class=\"token\">\u24d0<\/span>\\(\\left\\{\\left(-3,27\\right),\\left(-2,8\\right),\\left(-1,1\\right),\\left(0,0\\right),\\left(1,1\\right),\\left(2,8\\right),\\left(3,27\\right)\\right\\}\\) and <span class=\"token\">\u24d1<\/span> \\(\\left\\{\\left(0,0\\right),\\left(1,1\\right),\\left(4,2\\right),\\left(9,3\\right),\\left(16,4\\right)\\right\\}.\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<p id=\"fs-id1167836620263\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{ccccc}&amp; &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}\\left\\{\\left(-3,27\\right),\\left(-2,8\\right),\\left(-1,1\\right),\\left(0,0\\right),\\left(1,1\\right),\\left(2,8\\right),\\left(3,27\\right)\\right\\}\\hfill \\end{array}\\)\r\n\r\nEach <em data-effect=\"italics\">x<\/em>-value is matched with only one <em data-effect=\"italics\">y<\/em>-value. So this relation is a function.\r\n<p id=\"fs-id1167836684567\">But each <em data-effect=\"italics\">y<\/em>-value is not paired with only one <em data-effect=\"italics\">x<\/em>-value, \\(\\left(-3,27\\right)\\) and \\(\\left(3,27\\right),\\) for example. So this function is not one-to-one.<\/p>\r\n<p id=\"fs-id1167836531055\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{ccccc}&amp; &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}\\left\\{\\left(0,0\\right),\\left(1,1\\right),\\left(4,2\\right),\\left(9,3\\right),\\left(16,4\\right)\\right\\}\\hfill \\end{array}\\)\r\n\r\nEach <em data-effect=\"italics\">x<\/em>-value is matched with only one <em data-effect=\"italics\">y<\/em>-value. So this relation is a function.\r\n<p id=\"fs-id1167836546333\">Since each <em data-effect=\"italics\">y<\/em>-value is paired with only one <em data-effect=\"italics\">x<\/em>-value, this function is one-to-one.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829627366\">\r\n<div data-type=\"problem\" id=\"fs-id1167829695780\">\r\n<p id=\"fs-id1167836730111\">For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.<\/p>\r\n<p id=\"fs-id1167836689409\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\left(-3,-6\\right),\\left(-2,-4\\right),\\left(-1,-2\\right),\\left(0,0\\right),\\left(1,2\\right),\\left(2,4\\right),\\left(3,6\\right)\\right\\}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left\\{\\left(-4,8\\right),\\left(-2,4\\right),\\left(-1,2\\right),\\left(0,0\\right),\\left(1,2\\right),\\left(2,4\\right),\\left(4,8\\right)\\right\\}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836553296\">\r\n<p id=\"fs-id1167836558408\"><span class=\"token\">\u24d0<\/span> One-to-one function<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Function; not one-to-one\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836423521\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836389505\">\r\n<div data-type=\"problem\" id=\"fs-id1167829753177\">\r\n\r\nFor each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.\r\n\r\n<span class=\"token\">\u24d0<\/span>\\(\\left\\{\\left(27,-3\\right),\\left(8,-2\\right),\\left(1,-1\\right),\\left(0,0\\right),\\left(1,1\\right),\\left(8,2\\right),\\left(27,3\\right)\\right\\}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left\\{\\left(7,-3\\right),\\left(-5,-4\\right),\\left(8,0\\right),\\left(0,0\\right),\\left(-6,4\\right),\\left(-2,2\\right),\\left(-1,3\\right)\\right\\}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836628719\">\r\n\r\n<span class=\"token\">\u24d0<\/span> Not a function\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Function; not one-to-one\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836737936\">To help us determine whether a relation is a function, we use the <span data-type=\"term\" class=\"no-emphasis\">vertical line test<\/span>. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. Also, if any vertical line intersects the graph in more than one point, the graph does not represent a function.<\/p>\r\n<p id=\"fs-id1167836697122\">The vertical line is representing an <em data-effect=\"italics\">x<\/em>-value and we check that it intersects the graph in only one <em data-effect=\"italics\">y<\/em>-value. Then it is a function.<\/p>\r\n<p id=\"fs-id1167833369216\">To check if a function is one-to-one, we use a similar process. We use a horizontal line and check that each horizontal line intersects the graph in only one point. The horizontal line is representing a <em data-effect=\"italics\">y<\/em>-value and we check that it intersects the graph in only one <em data-effect=\"italics\">x<\/em>-value. If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function. This is the <span data-type=\"term\">horizontal line test<\/span>.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167829712822\">\r\n<div data-type=\"title\">Horizontal Line Test<\/div>\r\n<p id=\"fs-id1167836416189\">If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1168757697826\">We can test whether a graph of a relation is a function by using the vertical line test. We can then tell if the function is one-to-one by applying the horizontal line test.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167833257674\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829720214\">\r\n<div data-type=\"problem\" id=\"fs-id1167836692974\">\r\n<p id=\"fs-id1167836621109\">Determine <span class=\"token\">\u24d0<\/span> whether each graph is the graph of a function and, if so, <span class=\"token\">\u24d1<\/span> whether it is one-to-one.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836514026\" data-alt=\"This first graph shows a straight line passing through (0, 2) and (3, 0). This second shows a parabola opening up with vertex at (0, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_009_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This first graph shows a straight line passing through (0, 2) and (3, 0). This second shows a parabola opening up with vertex at (0, negative 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836685332\">\r\n<p id=\"fs-id1167836536170\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829693847\" data-alt=\"This figure shows a straight line passing through (0, 2) and (3, 0), with a red vertical line that only passes through one point and a blue horizontal line that only passes through one point.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_010_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line passing through (0, 2) and (3, 0), with a red vertical line that only passes through one point and a blue horizontal line that only passes through one point.\" \/><\/span>\r\n<p id=\"fs-id1167833224372\">Since any vertical line intersects the graph in at most one point, the graph is the graph of a function. Since any horizontal line intersects the graph in at most one point, the graph is the graph of a one-to-one function.<\/p>\r\n<p id=\"fs-id1167824739558\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829597115\" data-alt=\"This figure shows a parabola opening up with vertex at (0, negative 1), with a red vertical line that only passes through one point and a blue horizontal line that passes through two points.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_011_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening up with vertex at (0, negative 1), with a red vertical line that only passes through one point and a blue horizontal line that passes through two points.\" \/><\/span>\r\n<p id=\"fs-id1167836684122\">Since any vertical line intersects the graph in at most one point, the graph is the graph of a function. The horizontal line shown on the graph intersects it in two points. This graph does not represent a one-to-one function.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836610440\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836729138\">\r\n<div data-type=\"problem\" id=\"fs-id1167836529356\">\r\n<p id=\"fs-id1167836732672\">Determine <span class=\"token\">\u24d0<\/span> whether each graph is the graph of a function and, if so, <span class=\"token\">\u24d1<\/span> whether it is one-to-one.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167829688608\" data-alt=\"Graph a shows a parabola opening to the right with vertex at (negative 1, 0). Graph b shows an exponential function that does not cross the x axis and that passes through (0, 1) before increasing rapidly.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Graph a shows a parabola opening to the right with vertex at (negative 1, 0). Graph b shows an exponential function that does not cross the x axis and that passes through (0, 1) before increasing rapidly.\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829833822\">\r\n<p id=\"fs-id1167836630189\"><span class=\"token\">\u24d0<\/span> Not a function <span class=\"token\">\u24d1<\/span> One-to-one function<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167833327286\">\r\n<div data-type=\"problem\" id=\"fs-id1167829893597\">\r\n<p id=\"fs-id1167829787025\">Determine <span class=\"token\">\u24d0<\/span> whether each graph is the graph of a function and, if so, <span class=\"token\">\u24d1<\/span> whether it is one-to-one.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836622399\" data-alt=\"Graph a shows a parabola opening up with vertex at (0, 3). Graph b shows a straight line passing through (0, negative 2) and (2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_013_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Graph a shows a parabola opening up with vertex at (0, 3). Graph b shows a straight line passing through (0, negative 2) and (2, 0).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829599594\">\r\n<p id=\"fs-id1167829809865\"><span class=\"token\">\u24d0<\/span> Function; not one-to-one <span class=\"token\">\u24d1<\/span> One-to-one function<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836509386\">\r\n<h3 data-type=\"title\">Find the Inverse of a Function<\/h3>\r\n<p id=\"fs-id1167829784507\">Let\u2019s look at a one-to one function, \\(f\\), represented by the ordered pairs \\(\\left\\{\\left(0,5\\right),\\left(1,6\\right),\\left(2,7\\right),\\left(3,8\\right)\\right\\}.\\) For each \\(x\\)-value, \\(f\\) adds 5 to get the \\(y\\)-value. To \u2018undo\u2019 the addition of 5, we subtract 5 from each \\(y\\)-value and get back to the original \\(x\\)-value. We can call this \u201ctaking the inverse of \\(f\\)\u201d and name the function \\({f}^{-1}.\\)<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836330596\" data-alt=\"This figure shows the set (0, 5), (1, 6), (2, 7) and (3, 8) on the left side of an oval. The oval contains the numbers 0, 1, 2, and 3. There are black arrows from these numbers that point to the numbers 5, 6, 7, and 8, respectively in a second oval to the right of the first. Above this, there is a black arrow labeled \u201cf add 5\u201d coming from the left oval to the right oval. There are red arrows from the numbers 5, 6, 7, and 8 in the right oval to the numbers 0, 1, 2, and 3, respectively, in the left oval. Below this, we have a red arrow labeled \u201cf with a superscript negative 1\u201d and \u201csubtract 5\u201d. To the right of this, we have the set (5, 0), (6, 1), (7, 2) and (8, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_014_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the set (0, 5), (1, 6), (2, 7) and (3, 8) on the left side of an oval. The oval contains the numbers 0, 1, 2, and 3. There are black arrows from these numbers that point to the numbers 5, 6, 7, and 8, respectively in a second oval to the right of the first. Above this, there is a black arrow labeled \u201cf add 5\u201d coming from the left oval to the right oval. There are red arrows from the numbers 5, 6, 7, and 8 in the right oval to the numbers 0, 1, 2, and 3, respectively, in the left oval. Below this, we have a red arrow labeled \u201cf with a superscript negative 1\u201d and \u201csubtract 5\u201d. To the right of this, we have the set (5, 0), (6, 1), (7, 2) and (8, 3).\" \/><\/span>\r\n<p id=\"fs-id1167836619718\">Notice that that the ordered pairs of \\(f\\) and \\({f}^{-1}\\) have their \\(x\\)-values and \\(y\\)-values reversed. The domain of \\(f\\) is the range of \\({f}^{-1}\\) and the domain of \\({f}^{-1}\\) is the range of \\(f.\\)<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836493064\">\r\n<div data-type=\"title\">Inverse of a Function Defined by Ordered Pairs<\/div>\r\nIf \\(f\\left(x\\right)\\) is a one-to-one function whose ordered pairs are of the form \\(\\left(x,y\\right),\\) then its inverse function \\({f}^{-1}\\left(x\\right)\\) is the set of ordered pairs \\(\\left(y,x\\right).\\)\r\n\r\n<\/div>\r\n<p id=\"fs-id1167836700044\">In the next example we will find the inverse of a function defined by ordered pairs.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167836492441\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836537933\">\r\n<div data-type=\"problem\" id=\"fs-id1167829879570\">\r\n<p id=\"fs-id1167833007322\">Find the inverse of the function \\(\\left\\{\\left(0,3\\right),\\left(1,5\\right),\\left(2,7\\right),\\left(3,9\\right)\\right\\}.\\) Determine the domain and range of the inverse function.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<p id=\"fs-id1167836328806\">This function is one-to-one since every \\(x\\)-value is paired with exactly one \\(y\\)-value.<\/p>\r\n<p id=\"fs-id1167836409343\">To find the inverse we reverse the \\(x\\)-values and \\(y\\)-values in the ordered pairs of the function.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cccccc}\\text{Function}\\hfill &amp; &amp; &amp; &amp; &amp; \\left\\{\\left(0,3\\right),\\left(1,5\\right),\\left(2,7\\right),\\left(3,9\\right)\\right\\}\\hfill \\\\ \\text{Inverse Function}\\hfill &amp; &amp; &amp; &amp; &amp; \\left\\{\\left(3,0\\right),\\left(5,1\\right),\\left(7,2\\right),\\left(9,3\\right)\\right\\}\\hfill \\\\ \\text{Domain of Inverse Function}\\hfill &amp; &amp; &amp; &amp; &amp; \\left\\{3,5,7,9\\right\\}\\hfill \\\\ \\text{Range of Inverse Function}\\hfill &amp; &amp; &amp; &amp; &amp; \\left\\{0,1,2,3\\right\\}\\hfill \\end{array}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167824736277\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167824736280\">\r\n<div data-type=\"problem\" id=\"fs-id1167836531849\">\r\n<p id=\"fs-id1167836531851\">Find the inverse of \\(\\left\\{\\left(0,4\\right),\\left(1,7\\right),\\left(2,10\\right),\\left(3,13\\right)\\right\\}.\\) Determine the domain and range of the inverse function.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833135456\">\r\n<p id=\"fs-id1167833135458\">Inverse function: \\(\\left\\{\\left(4,0\\right),\\left(7,1\\right),\\left(10,2\\right),\\left(13,3\\right)\\right\\}.\\) Domain: \\(\\left\\{4,7,10,13\\right\\}.\\) Range: \\(\\left\\{0,1,2,3\\right\\}.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1171792512793\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1171790741157\">\r\n<div data-type=\"problem\" id=\"fs-id1171790275560\">\r\n<p id=\"fs-id1171792545899\">Find the inverse of \\(\\left\\{\\left(-1,4\\right),\\left(-2,1\\right),\\left(-3,0\\right),\\left(-4,2\\right)\\right\\}.\\) Determine the domain and range of the inverse function.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1171792843700\">\r\n<p id=\"fs-id1171790626716\">Inverse function: \\(\\left\\{\\left(4,-1\\right),\\left(1,-2\\right),\\left(0,-3\\right),\\left(2,-4\\right)\\right\\}.\\) Domain: \\(\\left\\{0,1,2,4\\right\\}.\\) Range: \\(\\left\\{-4,-3,-2,-1\\right\\}.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167833024456\">We just noted that if \\(f\\left(x\\right)\\) is a one-to-one function whose ordered pairs are of the form \\(\\left(x,y\\right),\\) then its inverse function \\({f}^{-1}\\left(x\\right)\\) is the set of ordered pairs \\(\\left(y,x\\right).\\)<\/p>\r\n<p id=\"fs-id1167832926088\">So if a point \\(\\left(a,b\\right)\\) is on the graph of a function \\(f\\left(x\\right),\\) then the ordered pair \\(\\left(b,a\\right)\\) is on the graph of \\({f}^{-1}\\left(x\\right).\\) See <a href=\"#CNX_IntAlg_Figure_10_01_015\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\r\n\r\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_10_01_015\"><span data-type=\"media\" id=\"fs-id1167829593818\" data-alt=\"This figure shows the line y equals x with points (3,1) and (1,3) on either side of the line. These two points are connected by a dashed blue line segment.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_015.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the line y equals x with points (3,1) and (1,3) on either side of the line. These two points are connected by a dashed blue line segment.\" \/><\/span><\/div>\r\n<p id=\"fs-id1167829609063\">The distance between any two pairs \\(\\left(a,b\\right)\\) and \\(\\left(b,a\\right)\\) is cut in half by the line \\(y=x.\\) So we say the points are mirror images of each other through the line \\(y=x.\\)<\/p>\r\n<p id=\"fs-id1167829746002\">Since every point on the graph of a function \\(f\\left(x\\right)\\) is a mirror image of a point on the graph of \\({f}^{-1}\\left(x\\right),\\) we say the graphs are mirror images of each other through the line \\(y=x.\\) We will use this concept to graph the inverse of a function in the next example.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167829620960\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829620962\">\r\n<div data-type=\"problem\" id=\"fs-id1167829620964\">\r\n\r\nGraph, on the same coordinate system, the inverse of the one-to one function shown.\r\n\r\n<span data-type=\"media\" id=\"fs-id1167829620970\" data-alt=\"This figure shows a line from (negative 5, negative 3) to (negative 3, negative 1) then to (negative 1,0) then to (0,2) and then to (3, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_016_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a line from (negative 5, negative 3) to (negative 3, negative 1) then to (negative 1,0) then to (0,2) and then to (3, 4).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836646061\">\r\n<p id=\"fs-id1167836646063\">We can use points on the graph to find points on the inverse graph. Some points on the graph are: \\(\\left(-5,-3\\right),\\left(-3,-1\\right),\\left(-1,0\\right),\\left(0,2\\right),\\left(3,4\\right)\\).<\/p>\r\n<p id=\"fs-id1167836532837\">So, the inverse function will contain the points: \\(\\left(-3,-5\\right),\\left(-1,-3\\right),\\left(0,-1\\right),\\left(2,0\\right),\\left(4,3\\right)\\).<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167832940223\" data-alt=\"This figure shows a line from (negative 5, negative 3) to (negative 3, negative 1) then to (negative 1, 0) then to (0,2) and then to (3, 4). Then there is a dashed line to denote y equals x. There is also a line from (negative 3, negative 5) to (negative 1, negative 3) then to (0, negative 1), then to (2, 0) and then to (4, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_017_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a line from (negative 5, negative 3) to (negative 3, negative 1) then to (negative 1, 0) then to (0,2) and then to (3, 4). Then there is a dashed line to denote y equals x. There is also a line from (negative 3, negative 5) to (negative 1, negative 3) then to (0, negative 1), then to (2, 0) and then to (4, 3).\" \/><\/span>\r\n<p id=\"fs-id1167832940234\">Notice how the graph of the original function and the graph of the inverse functions are mirror images through the line \\(y=x.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167826077493\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167826077496\">\r\n<div data-type=\"problem\" id=\"fs-id1167826077498\">\r\n<p id=\"fs-id1167826077500\">Graph, on the same coordinate system, the inverse of the one-to one function.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167826077504\" data-alt=\"The graph shows a line from (negative 3, negative 4) to (negative 2, negative 2) then to (0, negative 1), then to (1, 2) and then to (4, 3). The graph shows a line from (negative 3, 4) to (0, 3) then to (1, 2) and then to (4, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_018_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows a line from (negative 3, negative 4) to (negative 2, negative 2) then to (0, negative 1), then to (1, 2) and then to (4, 3). The graph shows a line from (negative 3, 4) to (0, 3) then to (1, 2) and then to (4, 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826077515\"><span data-type=\"media\" id=\"fs-id1167826077518\" data-alt=\"This figure shows a line from (negative 4, negative 3) to (negative 2, negative 2) then to (negative 1, 0) then to (2, 1) and then to (3, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_301_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a line from (negative 4, negative 3) to (negative 2, negative 2) then to (negative 1, 0) then to (2, 1) and then to (3, 4).\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836599086\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836599090\">\r\n<div data-type=\"problem\" id=\"fs-id1167836599092\">\r\n<p id=\"fs-id1167836599094\">Graph, on the same coordinate system, the inverse of the one-to one function.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1171790297071\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_022_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836599098\"><span data-type=\"media\" id=\"fs-id1167836599101\" data-alt=\"Graph extends from negative 4 to 4 on both axes. Points plotted are (negative 3, 4), (0, 3), (1, 2), and (4, 1). Line segments connect points.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Graph extends from negative 4 to 4 on both axes. Points plotted are (negative 3, 4), (0, 3), (1, 2), and (4, 1). Line segments connect points.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836625707\">When we began our discussion of an inverse function, we talked about how the inverse function \u2018undoes\u2019 what the original function did to a value in its domain in order to get back to the original <em data-effect=\"italics\">x<\/em>-value.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836625719\" data-alt=\"This figure shows x as the input to a box denoted as function f with f of x as the output of the box. Then, f of x is the input to a box denoted as function f superscript negative 1 with f superscript negative 1 of f of x equals x as the output of the box.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_019_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows x as the input to a box denoted as function f with f of x as the output of the box. Then, f of x is the input to a box denoted as function f superscript negative 1 with f superscript negative 1 of f of x equals x as the output of the box.\" \/><\/span>\r\n<div data-type=\"note\" id=\"fs-id1167836625730\">\r\n<div data-type=\"title\">Inverse Functions<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1167836625735\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill {f}^{-1}\\left(f\\left(x\\right)\\right)&amp; =\\hfill &amp; x,\\phantom{\\rule{0.2em}{0ex}}\\text{for all}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{in the domain of}\\phantom{\\rule{0.2em}{0ex}}f\\hfill \\\\ \\hfill f\\left({f}^{-1}\\left(x\\right)\\right)&amp; =\\hfill &amp; x,\\phantom{\\rule{0.2em}{0ex}}\\text{for all}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{in the domain of}\\phantom{\\rule{0.2em}{0ex}}{f}^{-1}\\hfill \\end{array}\\)<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829579603\">We can use this property to verify that two functions are inverses of each other.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167829579606\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829579608\">\r\n<div data-type=\"problem\" id=\"fs-id1167829579610\">\r\n<p id=\"fs-id1167829579612\">Verify that \\(f\\left(x\\right)=5x-1\\) and \\(g\\left(x\\right)=\\frac{x+1}{5}\\) are inverse functions.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836543935\">\r\n<p id=\"fs-id1167836543937\">The functions are inverses of each other if \\(g\\left(f\\left(x\\right)\\right)=x\\) and \\(f\\left(g\\left(x\\right)\\right)=x.\\)<\/p>\r\n\r\n<table id=\"fs-id1167829851289\" class=\"unnumbered unstyled\" summary=\"We want to examine whether g of f of x equals x In the first step, we substitute 5 x minus 1 for f of x which means that we are checking whether g of 5 x minus 1 equals x Then we find g of 5 x minus 1 where g of x equals the quantity x plus 1 all over 5. This means that we are trying to check whether the quantity 5 x plus 1 minus 1 all over 5 equals x. When we simplify we see that we are trying to check whether 5 x over 5 equals x. After simplifying further, we see that x equals x so this result holds. To investigate the other way, we are trying to determine whether f of g of x equals x. In the first step, we substitute the quantity x plus 1 all over 5 for g of x which means that we are trying to check whether f of quantity x plus 1 all over 5 equals x. Then we find f of quantity x plus 1 all over 5 where f of x equals 5 x minus 1. This means that we are trying to determine whether 5 times the quantity x plus 1 all over 5, minus 1 equals x. We simplify to see that to get x plus 1 minus 1 equals x. Simplifying further, we see that x does indeed equal x.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836524802\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute \\(5x-1\\) for \\(f\\left(x\\right).\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167825791236\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167825791245\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829715425\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829715444\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829696081\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Substitute \\(\\frac{x+1}{5}\\) for \\(g\\left(x\\right).\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829718205\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829718228\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833202354\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832999569\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167832999579\">Since both \\(g\\left(f\\left(x\\right)\\right)=x\\) and \\(f\\left(g\\left(x\\right)\\right)=x\\) are true, the functions \\(f\\left(x\\right)=5x-1\\) and \\(g\\left(x\\right)=\\frac{x+1}{5}\\) are inverse functions. That is, they are inverses of each other.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836399486\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836399489\">\r\n<div data-type=\"problem\" id=\"fs-id1167836399491\">\r\n<p id=\"fs-id1167836399493\">Verify that the functions are inverse functions.<\/p>\r\n<p id=\"fs-id1167836399496\">\\(f\\left(x\\right)=4x-3\\) and \\(g\\left(x\\right)=\\frac{x+3}{4}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833020781\">\r\n<p id=\"fs-id1167833020783\">\\(g\\left(f\\left(x\\right)\\right)=x,\\) and \\(f\\left(g\\left(x\\right)\\right)=x,\\) so they are inverses.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829715056\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829715060\">\r\n<div data-type=\"problem\" id=\"fs-id1167829715062\">\r\n<p id=\"fs-id1167829715064\">Verify that the functions are inverse functions.<\/p>\r\n<p id=\"fs-id1167829715067\">\\(f\\left(x\\right)=2x+6\\) and \\(g\\left(x\\right)=\\frac{x-6}{2}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829687013\">\r\n<p id=\"fs-id1167829687016\">\\(g\\left(f\\left(x\\right)\\right)=x,\\) and \\(f\\left(g\\left(x\\right)\\right)=x,\\) so they are inverses.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836333437\">We have found inverses of function defined by ordered pairs and from a graph. We will now look at how to find an inverse using an algebraic equation. The method uses the idea that if \\(f\\left(x\\right)\\) is a one-to-one function with ordered pairs \\(\\left(x,y\\right),\\) then its inverse function \\({f}^{-1}\\left(x\\right)\\) is the set of ordered pairs \\(\\left(y,x\\right).\\)<\/p>\r\n<p id=\"fs-id1167833345792\">If we reverse the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> in the function and then solve for <em data-effect=\"italics\">y<\/em>, we get our <span data-type=\"term\" class=\"no-emphasis\">inverse function<\/span>.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167833345812\" class=\"textbox textbox--examples\">\r\n<div data-type=\"title\">How to Find the inverse of a One-to-One Function<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833239447\">\r\n<div data-type=\"problem\" id=\"fs-id1167833239449\">\r\n<p id=\"fs-id1167833239451\">Find the inverse of \\(f\\left(x\\right)=4x+7.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167821865222\"><span data-type=\"media\" id=\"fs-id1167821865224\" data-alt=\"Step 1 is to substitute y for f of x. To do so, we replace f of x with y. Hence, f of x equals 4 x plus 7 becomes y equals 4 x plus 7.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to substitute y for f of x. To do so, we replace f of x with y. Hence, f of x equals 4 x plus 7 becomes y equals 4 x plus 7.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167821865234\" data-alt=\"Step 2 is to interchange the variables x and y. To do so, we replace x with y and then y with x. Hence, we obtain x equals 4y plus 7.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to interchange the variables x and y. To do so, we replace x with y and then y with x. Hence, we obtain x equals 4y plus 7.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167821865243\" data-alt=\"Step 3 is to solve for y. To do so, we subtract 7 from each side and then divide by 4. Hence, we have x minus 7 equals 4y and then the quantity x minus 7 divided by 4 equals y.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve for y. To do so, we subtract 7 from each side and then divide by 4. Hence, we have x minus 7 equals 4y and then the quantity x minus 7 divided by 4 equals y.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836495426\" data-alt=\"Step 4 is to substitute f superscript negative 1 of x for y. To do so, we replace y with f superscript negative 1 of x. Hence, the quantity x minus 7 divided by 4 equals f superscript negative 1 of x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to substitute f superscript negative 1 of x for y. To do so, we replace y with f superscript negative 1 of x. Hence, the quantity x minus 7 divided by 4 equals f superscript negative 1 of x.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836495436\" data-alt=\"Step 5 is to verify that the functions are inverses. To do so, we show that f superscript negative 1 of f of x equals x and that f of f superscript negative 1of x equals x. Hence, we ask whether f inverse of 4x plus 7 equals x. This becomes a question of whether 4 x plus 7 minus 7 all divided by 4 equals x. This becomes a question of whether 4x divided by 4 equals x. This is true. To show the other side, we examine whether f of f inverse of x equals x. This becomes a question of whether f of the quantity x minus 7 divided by 4 equals x. This becomes a question of whether 4 times the quantity x minus 7 divided by 4 equals x. This becomes a question of whether x minus 7 plus 7 equals x. This is true.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to verify that the functions are inverses. To do so, we show that f superscript negative 1 of f of x equals x and that f of f superscript negative 1of x equals x. Hence, we ask whether f inverse of 4x plus 7 equals x. This becomes a question of whether 4 x plus 7 minus 7 all divided by 4 equals x. This becomes a question of whether 4x divided by 4 equals x. This is true. To show the other side, we examine whether f of f inverse of x equals x. This becomes a question of whether f of the quantity x minus 7 divided by 4 equals x. This becomes a question of whether 4 times the quantity x minus 7 divided by 4 equals x. This becomes a question of whether x minus 7 plus 7 equals x. This is true.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836495454\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836495457\">\r\n<div data-type=\"problem\" id=\"fs-id1167833239525\">\r\n<p id=\"fs-id1167833239527\">Find the inverse of the function \\(f\\left(x\\right)=5x-3.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833239556\">\r\n<p id=\"fs-id1167833239558\">\\({f}^{-1}\\left(x\\right)=\\frac{x+3}{5}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829909640\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829909643\">\r\n<div data-type=\"problem\" id=\"fs-id1167829909645\">\r\n<p id=\"fs-id1167829909647\">Find the inverse of the function \\(f\\left(x\\right)=8x+5.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836537615\">\r\n<p id=\"fs-id1167836537617\">\\({f}^{-1}\\left(x\\right)=\\frac{x-5}{8}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829738768\">We summarize the steps below.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167829738771\" class=\"howto\">\r\n<div data-type=\"title\">How to Find the inverse of a One-to-One Function<\/div>\r\n<ol id=\"fs-id1167829738778\" class=\"stepwise\" type=\"1\">\r\n \t<li>Substitute <em data-effect=\"italics\">y<\/em> for \\(f\\left(x\\right).\\)<\/li>\r\n \t<li>Interchange the variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/li>\r\n \t<li>Solve for <em data-effect=\"italics\">y<\/em>.<\/li>\r\n \t<li>Substitute \\({f}^{-1}\\left(x\\right)\\) for <em data-effect=\"italics\">y<\/em>.<\/li>\r\n \t<li>Verify that the functions are inverses.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167829650658\" class=\"textbox textbox--examples\">\r\n<div data-type=\"title\">How to Find the Inverse of a One-to-One Function<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829650664\">\r\n<div data-type=\"problem\" id=\"fs-id1167833350434\">\r\n<p id=\"fs-id1167833350436\">Find the inverse of \\(f\\left(x\\right)=\\sqrt[5]{2x-3}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826172026\">\r\n<p id=\"fs-id1167826172028\">\\(\\begin{array}{cccccccc}&amp; &amp; &amp; &amp; &amp; \\hfill f\\left(x\\right)&amp; =\\hfill &amp; \\sqrt[5]{2x-3}\\hfill \\\\ \\text{Substitute}\\phantom{\\rule{0.2em}{0ex}}y\\phantom{\\rule{0.2em}{0ex}}\\text{for}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; \\sqrt[5]{2x-3}\\hfill \\\\ \\text{Interchange the variables}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}y.\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; \\sqrt[5]{2y-3}\\hfill \\\\ \\text{Solve for}\\phantom{\\rule{0.2em}{0ex}}y.\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill {\\left(x\\right)}^{5}&amp; =\\hfill &amp; {\\left(\\sqrt[5]{2y-3}\\right)}^{5}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {x}^{5}&amp; =\\hfill &amp; 2y-3\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {x}^{5}+3&amp; =\\hfill &amp; 2y\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{5}+3}{2}&amp; =\\hfill &amp; y\\hfill \\\\ \\text{Substitute}\\phantom{\\rule{0.2em}{0ex}}{f}^{-1}\\left(x\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{for}\\phantom{\\rule{0.2em}{0ex}}y.\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill {f}^{-1}\\left(x\\right)&amp; =\\hfill &amp; \\frac{{x}^{5}+3}{2}\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167836399012\">Verify that the functions are inverses.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{}\\\\ \\\\ \\hfill {f}^{-1}\\left(f\\left(x\\right)\\right)&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill f\\left({f}^{-1}\\left(x\\right)\\right)&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill \\\\ \\hfill {f}^{-1}\\left(\\sqrt[5]{2x-3}\\right)&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill f\\left(\\frac{{x}^{5}+3}{2}\\right)&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill \\\\ \\hfill \\frac{{\\left(\\sqrt[5]{2x-3}\\right)}^{5}+3}{2}&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\sqrt[5]{2\\left(\\frac{{x}^{5}+3}{2}\\right)-3}&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill \\\\ \\hfill \\frac{2x-3+3}{2}&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\sqrt[5]{{x}^{5}+3-3}&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill \\\\ \\hfill \\frac{2x}{2}&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\sqrt[5]{{x}^{5}}&amp; \\stackrel{?}{=}\\hfill &amp; x\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; x\u2713\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; x\u2713\\hfill \\end{array}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836480963\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836480966\">\r\n<div data-type=\"problem\" id=\"fs-id1167836480968\">\r\n<p id=\"fs-id1167836480970\">Find the inverse of the function \\(f\\left(x\\right)=\\sqrt[5]{3x-2}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829832059\">\r\n<p id=\"fs-id1167829832062\">\\({f}^{-1}\\left(x\\right)=\\frac{{x}^{5}+2}{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167833020910\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167833020913\">\r\n<div data-type=\"problem\" id=\"fs-id1167833020915\">\r\n<p id=\"fs-id1167833020917\">Find the inverse of the function \\(f\\left(x\\right)=\\sqrt[4]{6x-7}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836477385\">\r\n<p id=\"fs-id1167836477387\">\\({f}^{-1}\\left(x\\right)=\\frac{{x}^{4}+7}{6}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829906171\">\r\n<h3 data-type=\"title\">Key Concepts<\/h3>\r\n<ul id=\"fs-id1167829906177\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Composition of Functions:<\/strong> The composition of functions \\(f\\) and \\(g,\\) is written \\(f\\circ g\\) and is defined by\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"equation\" id=\"fs-id1167829879433\" class=\"unnumbered\" data-label=\"\">\\(\\left(f\\circ g\\right)\\left(x\\right)=f\\left(g\\left(x\\right)\\right)\\)<\/div>\r\n<div data-type=\"newline\"><\/div>\r\nWe read \\(f\\left(g\\left(x\\right)\\right)\\) as \\(f\\) of \\(g\\) of \\(x.\\)<\/li>\r\n \t<li><strong data-effect=\"bold\">Horizontal Line Test:<\/strong> If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.<\/li>\r\n \t<li><strong data-effect=\"bold\">Inverse of a Function Defined by Ordered Pairs:<\/strong> If \\(f\\left(x\\right)\\) is a one-to-one function whose ordered pairs are of the form \\(\\left(x,y\\right),\\) then its inverse function \\({f}^{-1}\\left(x\\right)\\) is the set of ordered pairs \\(\\left(y,x\\right).\\)<\/li>\r\n \t<li><strong data-effect=\"bold\">Inverse Functions:<\/strong> For every \\(x\\) in the domain of one-to-one function \\(f\\) and \\({f}^{-1},\\)\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"equation\" id=\"fs-id1167836544056\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill {f}^{-1}\\left(f\\left(x\\right)\\right)&amp; =\\hfill &amp; x\\hfill \\\\ \\hfill f\\left({f}^{-1}\\left(x\\right)\\right)&amp; =\\hfill &amp; x\\hfill \\end{array}\\)<\/div><\/li>\r\n \t<li><strong data-effect=\"bold\">How to Find the Inverse of a One-to-One Function:<\/strong>\r\n<ol id=\"fs-id1167836768376\" class=\"stepwise\" type=\"1\">\r\n \t<li>Substitute <em data-effect=\"italics\">y<\/em> for \\(f\\left(x\\right).\\)<\/li>\r\n \t<li>Interchange the variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/li>\r\n \t<li>Solve for <em data-effect=\"italics\">y<\/em>.<\/li>\r\n \t<li>Substitute \\({f}^{-1}\\left(x\\right)\\) for \\(y.\\)<\/li>\r\n \t<li>Verify that the functions are inverses.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833020689\">\r\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167833020692\">\r\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\r\n<p id=\"fs-id1167833020699\"><strong data-effect=\"bold\">Find and Evaluate Composite Functions<\/strong><\/p>\r\n<p id=\"fs-id1167833020704\">In the following exercises, find <span class=\"token\">\u24d0<\/span> (<em data-effect=\"italics\">f<\/em> \u2218 <em data-effect=\"italics\">g<\/em>)(<em data-effect=\"italics\">x<\/em>), <span class=\"token\">\u24d1<\/span> (<em data-effect=\"italics\">g<\/em> \u2218 <em data-effect=\"italics\">f<\/em>)(<em data-effect=\"italics\">x<\/em>), and <span class=\"token\">\u24d2<\/span> (<em data-effect=\"italics\">f<\/em> \u00b7 <em data-effect=\"italics\">g<\/em>)(<em data-effect=\"italics\">x<\/em>).<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829721059\" class=\"material-set-2\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167829721063\">\\(f\\left(x\\right)=4x+3\\) and \\(g\\left(x\\right)=2x+5\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829650613\">\r\n<p id=\"fs-id1167829650615\"><span class=\"token\">\u24d0<\/span>\\(8x+23\\)<span class=\"token\">\u24d1<\/span>\\(8x+11\\)<span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(8{x}^{2}+26x+15\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167825708408\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167825708410\">\r\n<p id=\"fs-id1167825708412\">\\(f\\left(x\\right)=3x-1\\) and \\(g\\left(x\\right)=5x-3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826025403\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167826025405\">\r\n<p id=\"fs-id1167826025407\">\\(f\\left(x\\right)=6x-5\\) and \\(g\\left(x\\right)=4x+1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836790650\">\r\n<p id=\"fs-id1167836790652\"><span class=\"token\">\u24d0<\/span>\\(24x+1\\)<span class=\"token\">\u24d1<\/span>\\(24x-19\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(24{x}^{2}+19x-5\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829878007\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829878009\">\r\n<p id=\"fs-id1167829878011\">\\(f\\left(x\\right)=2x+7\\) and \\(g\\left(x\\right)=3x-4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829696400\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829696402\">\r\n<p id=\"fs-id1167829696404\">\\(f\\left(x\\right)=3x\\) and \\(g\\left(x\\right)=2{x}^{2}-3x\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829830707\">\r\n<p id=\"fs-id1167829830709\"><span class=\"token\">\u24d0<\/span>\\(6{x}^{2}-9x\\)<span class=\"token\">\u24d1<\/span>\\(18{x}^{2}-9x\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(6{x}^{3}-9{x}^{2}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829840934\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829840936\">\r\n<p id=\"fs-id1167829840938\">\\(f\\left(x\\right)=2x\\) and \\(g\\left(x\\right)=3{x}^{2}-1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836626117\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836626119\">\r\n<p id=\"fs-id1167836626121\">\\(f\\left(x\\right)=2x-1\\) and \\(g\\left(x\\right)={x}^{2}+2\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167825872598\">\r\n<p id=\"fs-id1167825872600\"><span class=\"token\">\u24d0<\/span>\\(2{x}^{2}+3\\)<span class=\"token\">\u24d1<\/span>\\(4{x}^{2}-4x+3\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(2{x}^{3}-{x}^{2}+4x-2\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167822996884\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167822996887\">\r\n<p id=\"fs-id1167822996889\">\\(f\\left(x\\right)=4x+3\\) and \\(g\\left(x\\right)={x}^{2}-4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836602658\">In the following exercises, find the values described.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836602662\" class=\"material-set-2\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167836602666\">For functions \\(f\\left(x\\right)=2{x}^{2}+3\\) and \\(g\\left(x\\right)=5x-1,\\) find<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(-2\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(-3\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> \\(\\left(f\\circ f\\right)\\left(-1\\right)\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833138071\">\r\n<p id=\"fs-id1167833138073\"><span class=\"token\">\u24d0<\/span> 245 <span class=\"token\">\u24d1<\/span> 104 <span class=\"token\">\u24d2<\/span> 53<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836607003\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836607006\">\r\n<p id=\"fs-id1167836607008\">For functions \\(f\\left(x\\right)=5{x}^{2}-1\\) and \\(g\\left(x\\right)=4x-1,\\) find<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(1\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(-1\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> \\(\\left(f\\circ f\\right)\\left(2\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829807251\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829807254\">\r\n<p id=\"fs-id1167829807256\">For functions \\(f\\left(x\\right)=2{x}^{3}\\) and \\(g\\left(x\\right)=3{x}^{2}+2,\\) find<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(-1\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(1\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> \\(\\left(g\\circ g\\right)\\left(1\\right)\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829833766\">\r\n<p id=\"fs-id1167829833768\"><span class=\"token\">\u24d0<\/span> 250 <span class=\"token\">\u24d1<\/span> 14 <span class=\"token\">\u24d2<\/span> 77<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836690124\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836690126\">\r\n<p id=\"fs-id1167836690128\">For functions \\(f\\left(x\\right)=3{x}^{3}+1\\) and \\(g\\left(x\\right)=2{x}^{2}-3,\\) find<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span> \\(\\left(f\\circ g\\right)\\left(-2\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> \\(\\left(g\\circ f\\right)\\left(-1\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> \\(\\left(g\\circ g\\right)\\left(1\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167825808240\"><strong data-effect=\"bold\">Determine Whether a Function is One-to-One<\/strong><\/p>\r\n<p id=\"fs-id1167825808246\">In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167825808250\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167825808252\">\r\n<p id=\"fs-id1167825808254\">\\(\\left\\{\\left(-3,9\\right),\\left(-2,4\\right),\\left(-1,1\\right),\\left(0,0\\right)\\),<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\left(1,1\\right),\\left(2,4\\right),\\left(3,9\\right)\\right\\}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836530984\">\r\n<p id=\"fs-id1167824668944\">Function; not one-to-one<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824668950\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167824668952\">\r\n<p id=\"fs-id1167824668954\">\\(\\left\\{\\left(9,-3\\right),\\left(4,-2\\right),\\left(1,-1\\right),\\left(0,0\\right)\\),<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\left(1,1\\right),\\left(4,2\\right),\\left(9,3\\right)\\right\\}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833369697\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167833369699\">\r\n<p id=\"fs-id1167833369701\">\\(\\left\\{\\left(-3,-5\\right),\\left(-2,-3\\right),\\left(-1,-1\\right)\\),<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\left(0,1\\right),\\left(1,3\\right),\\left(2,5\\right),\\left(3,7\\right)\\right\\}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167824704165\">\r\n<p id=\"fs-id1167824704167\">One-to-one function<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824704172\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167824704174\">\r\n<p id=\"fs-id1167824704177\">\\(\\left\\{\\left(5,3\\right),\\left(4,2\\right),\\left(3,1\\right),\\left(2,0\\right)\\),<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\left(1,-1\\right),\\left(0,-2\\right),\\left(-1,-3\\right)\\right\\}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836533576\">In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836533580\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836533582\">\r\n<p id=\"fs-id1167836533584\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829787690\" data-alt=\"This figure shows a graph of a circle with center at the origin and radius 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a graph of a circle with center at the origin and radius 3.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829787704\" data-alt=\"This figure shows a graph of a parabola opening upward with vertex at (0k, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_202_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a graph of a parabola opening upward with vertex at (0k, 2).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829787714\">\r\n<p id=\"fs-id1167829787716\"><span class=\"token\">\u24d0<\/span> Not a function <span class=\"token\">\u24d1<\/span> Function; not one-to-one<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829787729\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829787731\">\r\n<p id=\"fs-id1167829871910\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829871917\" data-alt=\"This figure shows a parabola opening to the right with vertex at (negative 2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_203_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening to the right with vertex at (negative 2, 0).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829871932\" data-alt=\"This figure shows a graph of a polynomial with odd order, so that it starts in the third quadrant, increases to the origin and then continues increasing through the first quadrant.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_204_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a graph of a polynomial with odd order, so that it starts in the third quadrant, increases to the origin and then continues increasing through the first quadrant.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167823012125\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167823012127\">\r\n<p id=\"fs-id1167823012129\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167823012136\" data-alt=\"This figure shows a graph of a curve that starts at (negative 6 negative 2) increases to the origin and then continues increasing slowly to (6, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_205_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a graph of a curve that starts at (negative 6 negative 2) increases to the origin and then continues increasing slowly to (6, 2).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167823012152\" data-alt=\"This figure shows a parabola opening upward with vertex at (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_206_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening upward with vertex at (0, negative 4).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167823012162\">\r\n<p id=\"fs-id1167823012164\"><span class=\"token\">\u24d0<\/span> One-to-one function<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Function; not one-to-one\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829701984\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829701986\">\r\n<p id=\"fs-id1167829701988\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829701994\" data-alt=\"This figure shows a straight line segment decreasing from (negative 4, 6) to (2, 0), after which it increases from (2, 0) to (6, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_207_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line segment decreasing from (negative 4, 6) to (2, 0), after which it increases from (2, 0) to (6, 4).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829702010\" data-alt=\"This figure shows a circle with radius 4 and center at the origin.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_208_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a circle with radius 4 and center at the origin.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836624981\">In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836624985\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836624987\">\r\n<p id=\"fs-id1167836624990\">\\(\\left\\{\\left(2,1\\right),\\left(4,2\\right),\\left(6,3\\right),\\left(8,4\\right)\\right\\}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829808255\">\r\n<p id=\"fs-id1167829808257\">Inverse function: \\(\\left\\{\\left(1,2\\right),\\left(2,4\\right),\\left(3,6\\right),\\left(4,8\\right)\\right\\}.\\) Domain: \\(\\left\\{1,2,3,4\\right\\}.\\) Range: \\(\\left\\{2,4,6,8\\right\\}.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836664617\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836664619\">\r\n<p id=\"fs-id1167829732032\">\\(\\left\\{\\left(6,2\\right),\\left(9,5\\right),\\left(12,8\\right),\\left(15,11\\right)\\right\\}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167825703072\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167825703074\">\r\n<p id=\"fs-id1167825703077\">\\(\\left\\{\\left(0,-2\\right),\\left(1,3\\right),\\left(2,7\\right),\\left(3,12\\right)\\right\\}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829890422\">\r\n<p id=\"fs-id1167829890424\">Inverse function: \\(\\left\\{\\left(-2,0\\right),\\left(3,1\\right),\\left(7,2\\right),\\left(12,3\\right)\\right\\}.\\) Domain: \\(\\left\\{-2,3,7,12\\right\\}.\\) Range: \\(\\left\\{0,1,2,3\\right\\}.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829609019\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829609022\">\r\n<p id=\"fs-id1167829609024\">\\(\\left\\{\\left(0,0\\right),\\left(1,1\\right),\\left(2,4\\right),\\left(3,9\\right)\\right\\}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167830019130\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829807304\">\r\n<p id=\"fs-id1167829807306\">\\(\\left\\{\\left(-2,-3\\right),\\left(-1,-1\\right),\\left(0,1\\right),\\left(1,3\\right)\\right\\}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832982002\">\r\n<p id=\"fs-id1167832982004\">Inverse function: \\(\\left\\{\\left(-3,\\text{\u2212}2\\right),\\left(-1,-1\\right),\\left(1,0\\right),\\left(3,1\\right)\\right\\}.\\) Domain: \\(\\left\\{-3,\\text{\u2212}1,1,3\\right\\}.\\) Range: \\(\\left\\{-2,-1,0,1\\right\\}.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829957984\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829957986\">\r\n<p id=\"fs-id1167829957988\">\\(\\left\\{\\left(5,3\\right),\\left(4,2\\right),\\left(3,1\\right),\\left(2,0\\right)\\right\\}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829690781\">In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829690786\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829690788\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829690791\" data-alt=\"This figure shows a series of line segments from (negative 4, negative 3) to (negative 3, 0) then to (negative 1, 2) and then to (3, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_209_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 4, negative 3) to (negative 3, 0) then to (negative 1, 2) and then to (3, 4).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829690802\"><span data-type=\"media\" id=\"fs-id1167829690805\" data-alt=\"This figure shows a series of line segments from (negative 3, negative 4) to (0, negative 3) then to (2, negative 1), and then to (4, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 3, negative 4) to (0, negative 3) then to (2, negative 1), and then to (4, 3).\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824732135\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167824732137\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167824732140\" data-alt=\"This figure shows a series of line segments from (negative 4, negative 4) to (negative 3, 1) then to (0, 2) and then to (2, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_210_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 4, negative 4) to (negative 3, 1) then to (0, 2) and then to (2, 4).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824732166\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167824732168\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167824732171\" data-alt=\"This figure shows a series of line segments from (negative 4, 4) to (0, 3) then to (3, 2) and then to (4, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_211_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 4, 4) to (0, 3) then to (3, 2) and then to (4, negative 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829840824\"><span data-type=\"media\" id=\"fs-id1167829840828\" data-alt=\"This figure shows a series of line segments from (negative 1, 4) to (2, 3) then to (3, 0), and then to (4, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 1, 4) to (2, 3) then to (3, 0), and then to (4, negative 4).\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829840839\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829840841\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829840845\" data-alt=\"This figure shows a series of line segments from (negative 4, negative 4) to (negative 1, negative 3) then to (0, 1), then to (1, 3), and then to (4, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_212_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 4, negative 4) to (negative 1, negative 3) then to (0, 1), then to (1, 3), and then to (4, 4).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836523852\">In the following exercises, determine whether or not the given functions are inverses.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836523856\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836523858\">\r\n<p id=\"fs-id1167836523860\">\\(f\\left(x\\right)=x+8\\) and \\(g\\left(x\\right)=x-8\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167824733031\">\r\n<p id=\"fs-id1167824733033\">\\(g\\left(f\\left(x\\right)\\right)=x,\\) and \\(f\\left(g\\left(x\\right)\\right)=x,\\) so they are inverses.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832936881\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167832936883\">\r\n<p id=\"fs-id1167832936885\">\\(f\\left(x\\right)=x-9\\) and \\(g\\left(x\\right)=x+9\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832926058\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167832926061\">\r\n<p id=\"fs-id1167832926063\">\\(f\\left(x\\right)=7x\\) and \\(g\\left(x\\right)=\\frac{x}{7}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829787270\">\r\n<p id=\"fs-id1167829787272\">\\(g\\left(f\\left(x\\right)\\right)=x,\\) and \\(f\\left(g\\left(x\\right)\\right)=x,\\) so they are inverses.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167825003602\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167825003604\">\r\n<p id=\"fs-id1167825003606\">\\(f\\left(x\\right)=\\frac{x}{11}\\) and \\(g\\left(x\\right)=11x\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829808958\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829808960\">\r\n<p id=\"fs-id1167829808962\">\\(f\\left(x\\right)=7x+3\\) and \\(g\\left(x\\right)=\\frac{x-3}{7}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836440128\">\r\n<p id=\"fs-id1167836440130\">\\(g\\left(f\\left(x\\right)\\right)=x,\\) and \\(f\\left(g\\left(x\\right)\\right)=x,\\) so they are inverses.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824726024\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829832983\">\r\n<p id=\"fs-id1167829832985\">\\(f\\left(x\\right)=5x-4\\) and \\(g\\left(x\\right)=\\frac{x-4}{5}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824765407\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167824765409\">\r\n<p id=\"fs-id1167824765411\">\\(f\\left(x\\right)=\\sqrt{x+2}\\) and \\(g\\left(x\\right)={x}^{2}-2\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833051870\">\r\n<p id=\"fs-id1167833051872\">\\(g\\left(f\\left(x\\right)\\right)=x,\\) and \\(f\\left(g\\left(x\\right)\\right)=x,\\) so they are inverses (for nonnegative \\(x\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829849326\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829849328\">\r\n<p id=\"fs-id1167829849330\">\\(f\\left(x\\right)=\\sqrt[3]{x-4}\\) and \\(g\\left(x\\right)={x}^{3}+4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167833022953\">In the following exercises, find the inverse of each function.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167833022957\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167833022959\">\r\n<p id=\"fs-id1167833022961\">\\(f\\left(x\\right)=x-12\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833022984\">\r\n<p id=\"fs-id1167833022986\">\\({f}^{-1}\\left(x\\right)=x+12\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" class=\"material-set-2\">\r\n<div data-type=\"problem\">\r\n\r\n\\(f\\left(x\\right)=x+17\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829859344\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829859346\">\r\n<p id=\"fs-id1167829859348\">\\(f\\left(x\\right)=9x\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167824602071\">\r\n<p id=\"fs-id1167824602073\">\\({f}^{-1}\\left(x\\right)=\\frac{x}{9}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824602102\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167824602104\">\r\n<p id=\"fs-id1167824602106\">\\(f\\left(x\\right)=8x\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829784467\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829784470\">\r\n<p id=\"fs-id1167829784472\">\\(f\\left(x\\right)=\\frac{x}{6}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829620578\">\r\n<p id=\"fs-id1167829620580\">\\({f}^{-1}\\left(x\\right)=6x\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829620608\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829620610\">\r\n<p id=\"fs-id1167829620612\">\\(f\\left(x\\right)=\\frac{x}{4}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836621636\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836621638\">\r\n<p id=\"fs-id1167836621640\">\\(f\\left(x\\right)=6x-7\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829811511\">\r\n<p id=\"fs-id1167829811513\">\\({f}^{-1}\\left(x\\right)=\\frac{x+7}{6}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829811547\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829811549\">\r\n<p id=\"fs-id1167829811552\">\\(f\\left(x\\right)=7x-1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829801847\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829801849\">\r\n<p id=\"fs-id1167829801851\">\\(f\\left(x\\right)=-2x+5\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833175196\">\r\n<p id=\"fs-id1167833175199\">\\({f}^{-1}\\left(x\\right)=\\frac{x-5}{-2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829715177\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829715179\">\r\n<p id=\"fs-id1167829715182\">\\(f\\left(x\\right)=-5x-4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167825003647\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167825003650\">\r\n<p id=\"fs-id1167825003652\">\\(f\\left(x\\right)={x}^{2}+6,\\)\\(x\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167825003690\">\r\n<p id=\"fs-id1167825003692\">\\({f}^{-1}\\left(x\\right)=\\sqrt{x-6}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826122987\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167826122990\">\r\n<p id=\"fs-id1167826122992\">\\(f\\left(x\\right)={x}^{2}-9,\\)\\(x\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829851254\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829851256\">\r\n<p id=\"fs-id1167829851258\">\\(f\\left(x\\right)={x}^{3}-4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829651284\">\r\n<p id=\"fs-id1167829651286\">\\({f}^{-1}\\left(x\\right)=\\sqrt[3]{x+4}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829651320\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829651322\">\r\n\r\n\\(f\\left(x\\right)={x}^{3}+6\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836575720\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836575722\">\r\n<p id=\"fs-id1167836575724\">\\(f\\left(x\\right)=\\frac{1}{x+2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836575752\">\r\n<p id=\"fs-id1167836575754\">\\({f}^{-1}\\left(x\\right)=\\frac{1}{x}-2\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836495478\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836495480\">\r\n<p id=\"fs-id1167836495483\">\\(f\\left(x\\right)=\\frac{1}{x-6}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829704844\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829704847\">\r\n<p id=\"fs-id1167829704849\">\\(f\\left(x\\right)=\\sqrt{x-2},\\)\\(x\\ge 2\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829704886\">\r\n<p id=\"fs-id1167829704888\">\\({f}^{-1}\\left(x\\right)={x}^{2}+2\\), \\(x\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167821885714\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167821885716\">\r\n<p id=\"fs-id1167821885718\">\\(f\\left(x\\right)=\\sqrt{x+8},\\)\\(x\\ge -8\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836494143\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836494145\">\r\n<p id=\"fs-id1167836494147\">\\(f\\left(x\\right)=\\sqrt[3]{x-3}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836494175\">\r\n<p id=\"fs-id1167836494177\">\\({f}^{-1}\\left(x\\right)={x}^{3}+3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833084919\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167833084921\">\r\n<p id=\"fs-id1167833084924\">\\(f\\left(x\\right)=\\sqrt[3]{x+5}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824917078\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167824917080\">\r\n<p id=\"fs-id1167824917082\">\\(f\\left(x\\right)=\\sqrt[4]{9x-5},\\)\\(x\\ge \\frac{5}{9}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836450145\">\r\n<p id=\"fs-id1167836450147\">\\({f}^{-1}\\left(x\\right)=\\frac{{x}^{4}+5}{9}\\), \\(x\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836450194\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836450196\">\r\n<p id=\"fs-id1167836450198\">\\(f\\left(x\\right)=\\sqrt[4]{8x-3},\\)\\(x\\ge \\frac{3}{8}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829808436\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829808438\">\r\n<p id=\"fs-id1167829808440\">\\(f\\left(x\\right)=\\sqrt[5]{-3x+5}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829808470\">\r\n<p id=\"fs-id1167829808472\">\\({f}^{-1}\\left(x\\right)=\\frac{{x}^{5}-5}{-3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836480837\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836480839\">\r\n<p id=\"fs-id1167836480841\">\\(f\\left(x\\right)=\\sqrt[5]{-4x-3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836514906\">\r\n<h4 data-type=\"title\">Writing Exercises<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1167836514914\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836514916\">\r\n<p id=\"fs-id1167836514918\">Explain how the graph of the inverse of a function is related to the graph of the function.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836514923\">\r\n<p id=\"fs-id1167829751136\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829751141\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829751144\">\r\n<p id=\"fs-id1167829751146\">Explain how to find the inverse of a function from its equation. Use an example to demonstrate the steps.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\">\r\n<h4 data-type=\"title\">Self Check<\/h4>\r\n<p id=\"fs-id1167829751165\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167829751177\" data-alt=\"This table has four rows and four columns. The first row, which serves as a header, reads I can\u2026, Confidently, With some help, and No\u2014I don\u2019t get it. The first column below the header row reads Find and evaluate composite functions, determine whether a function is one-to-one, and find the inverse of a function. The rest of the cells are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_213_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four rows and four columns. The first row, which serves as a header, reads I can\u2026, Confidently, With some help, and No\u2014I don\u2019t get it. The first column below the header row reads Find and evaluate composite functions, determine whether a function is one-to-one, and find the inverse of a function. The rest of the cells are blank.\" \/><\/span>\r\n<p id=\"fs-id1167829751188\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p>\r\n<p id=\"fs-id1167829751196\">\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p>\r\n<p id=\"fs-id1167836399239\">\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p>\r\n<p id=\"fs-id1167836399250\">\u2026no\u2014I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"glossary\" class=\"textbox shaded\">\r\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\r\n<dl>\r\n \t<dt>one-to-one function<\/dt>\r\n \t<dd id=\"fs-id1167836399269\">A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each <em data-effect=\"italics\">y<\/em>-value is matched with only one <em data-effect=\"italics\">x<\/em>-value.<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Find and evaluate composite functions<\/li>\n<li>Determine whether a function is one-to-one<\/li>\n<li>Find the inverse of a function<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836288458\" class=\"be-prepared\">\n<p id=\"fs-id1167836595563\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167836440251\" type=\"1\">\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5eaf591c6b46c0f97d5b0802f0751675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23a620bb08098b9f549a7844f620f40a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/> find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2882d38c9fbadedc61b2a3926c4bde9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/5e548626-8f0f-496d-ab87-4f0358ca2fd3#fs-id1167836521479\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c761066c8f4debb53d6f34ef44a0a9d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#43;&#50;&#121;&#61;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/b03538a1-8a7b-4158-a68b-e0e8a24c9fd4#fs-id1167835229496\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10862f86746b1fb368cd282a6d5f5a85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#53;&#125;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"83\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836390100\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167829579375\">In this chapter, we will introduce two new types of functions, exponential functions and logarithmic functions. These functions are used extensively in business and the sciences as we will see.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Find and Evaluate Composite Functions<\/h3>\n<p id=\"fs-id1167829747783\">Before we introduce the functions, we need to look at another operation on functions called <span data-type=\"term\" class=\"no-emphasis\">composition<\/span>. In composition, the output of one function is the input of a second function. For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa307d44fe899099cad9fc84395f6eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/> the composition is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ca4cf2da3c03e0d5a112084842d8569_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"36\" style=\"vertical-align: -4px;\" \/> and is defined by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af8ed387a36743cb01e4a46029c04236_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"170\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>We read <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-126f21dc35931f9b8d72c325f659566b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd3c682fce4fabf46641c71c34c46515_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#96;&#96;&#125;&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -4px;\" \/> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dbf889f754219a189c9c333c1b0a3fbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#39;&#39;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836294921\" data-alt=\"This figure shows x as the input to a box denoted as function g with g of x as the output of the box. Then, g of x is the input to a box denoted as function f with f of g of x as the output of the box.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows x as the input to a box denoted as function g with g of x as the output of the box. Then, g of x is the input to a box denoted as function f with f of g of x as the output of the box.\" \/><\/span><\/p>\n<p id=\"fs-id1167833212955\">To do a composition, the output of the first function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-344231958daaa52c1755b0b6892d4016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/> becomes the input of the second function, <em data-effect=\"italics\">f<\/em>, and so we must be sure that it is part of the domain of <em data-effect=\"italics\">f<\/em>.<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\">Composition of Functions<\/div>\n<p id=\"fs-id1167836417281\">The composition of functions <em data-effect=\"italics\">f<\/em> and <em data-effect=\"italics\">g<\/em> is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-304fd3c603fd213df7508bdcc85514c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&middot;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" style=\"vertical-align: -4px;\" \/> and is defined by<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829930414\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93ad33b10518db26515730aecb3748be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"162\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167829791406\">We read <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-126f21dc35931f9b8d72c325f659566b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> of <em data-effect=\"italics\">x<\/em>.<\/p>\n<\/div>\n<p id=\"fs-id1167836628588\">We have actually used composition without using the notation many times before. When we graphed quadratic functions using translations, we were composing functions. For example, if we first graphed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d55ceaf8ea5956f70e4c1ac4cb7eff90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -4px;\" \/> as a parabola and then shifted it down vertically four units, we were using the composition defined by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93ad33b10518db26515730aecb3748be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"162\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-937d82e8b6e34249e45ad67251a8e9f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167826024537\" data-alt=\"This figure shows x as the input to a box denoted as g of x equals x squared with x squared as the output of the box. Then, x squared is the input to a box denoted as f of x equals x minus 4 with f of g of x equals x squared minus 4 as the output of the box.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows x as the input to a box denoted as g of x equals x squared with x squared as the output of the box. Then, x squared is the input to a box denoted as f of x equals x minus 4 with f of g of x equals x squared minus 4 as the output of the box.\" \/><\/span><\/p>\n<p id=\"fs-id1167829851372\">The next example will demonstrate that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0b52a6a7e79ad20e28b25273f0d70a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ccbb80181856bf5404a5daac8ae4443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff77fc6d8ce39e0c137debd709c838d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -4px;\" \/> usually result in different outputs.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829718921\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836628343\">\n<div data-type=\"problem\" id=\"fs-id1167824781200\">\n<p id=\"fs-id1167836409476\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7574e71d26cef7230fafa54b13f2e16c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17a8592a058312909820ca18a86315e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#43;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> find: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0b52a6a7e79ad20e28b25273f0d70a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b73212161a55046821b0266955fa89d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-876d9e8ceacf4f1109e1fa956675d543_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836685094\">\n<p id=\"fs-id1167829860686\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836449256\" class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of f of g of x to obtain that f of g of x equals f of g of x. In the second step, we substitute 2 x plus 3 for g of x. This means that f of g of x equals f of 2 x plus 3. In the third step, we find f of 2 x plus 3 where f of x equals 4 x minus 5. This means that f of g of x equals 4 times the quantity 2 x plus 3, minus 5. In the fourth step, we distribute. This means that f of g of x equals 8 x plus 12 minus 5. In the fifth step, we simplify. This means that f of g of x equals 8 x plus 7.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the definition of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf80feb465e5ed34b951f57bd767bccb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829850078\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836507880\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836714410\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829787748\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836424078\" class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of g of f of x to obtain that g of f of x equals g of f of x. In the second step, we substitute 4 x minus 5 for f of x. This means that g of f of x equals g of 4 x minus 5. In the third step, we find g of 4 x minus 5 where g of x equals 2 x plus 3. This means that g of f of x equals 2 times the quantity 4 x minus 5, plus 3. In the fourth step, we distribute. This means that g of f of x equals 8 x minus 10 plus 3. In the fifth step, we simplify. This means that g of f of x equals 8 x minus 7.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the definition of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf80feb465e5ed34b951f57bd767bccb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832999727\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829907616\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824763273\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836485988\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836417376\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829620750\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832940195\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167833136344\">Notice the difference in the result in part <span class=\"token\">\u24d0<\/span> and part <span class=\"token\">\u24d1<\/span>.<\/p>\n<p><span class=\"token\">\u24d2<\/span> Notice that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff77fc6d8ce39e0c137debd709c838d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -4px;\" \/> is different than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf80feb465e5ed34b951f57bd767bccb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> In part <span class=\"token\">\u24d0<\/span> we did the composition of the functions. Now in part <span class=\"token\">\u24d2<\/span> we are not composing them, we are multiplying them.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c97328cbfd57aeae82644f423cc06aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#102;&#105;&#110;&#105;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#45;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#43;&#51;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"641\" style=\"vertical-align: -26px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836613359\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836576079\">\n<div data-type=\"problem\" id=\"fs-id1167829650480\">\n<p>For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48e26e890f6353b930c8bd0c3b43762e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf6d30c38be37a3126a9d13844da8864_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#120;&#43;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0d67da21b3684a96de1197797a460ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ccbb80181856bf5404a5daac8ae4443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff77fc6d8ce39e0c137debd709c838d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836699034\">\n<p id=\"fs-id1167833047859\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52581d01b7c7ff7e7318b4dd0b589ad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14c7db5cf5f055bfc8aa89394a51702f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce0d9b213e0c8f9e4766ec30d2846d26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824590525\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167824735112\">\n<p id=\"fs-id1167836299504\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa29e2e5428d664c8c762e8419ed12a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bf1c18217f4075469cf5bac0c2d2374_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#120;&#45;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0b52a6a7e79ad20e28b25273f0d70a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b73212161a55046821b0266955fa89d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-876d9e8ceacf4f1109e1fa956675d543_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&middot;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836447318\">\n<p id=\"fs-id1167836319241\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf21ed5729c5c14b7e4401319e3d7dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#120;&#45;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"68\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf21ed5729c5c14b7e4401319e3d7dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#120;&#45;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"68\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be260244865555ffddce4ee9e2af0dbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#56;&#120;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"124\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example we will evaluate a composition for a specific value.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833086943\">\n<p>For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f5bf49fa23646fea2903fec2cf5c09e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58fef4f19214c932a255283000f96cda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#43;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> find: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a79a2939b364798102ab7fa859fdcf13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29e2171183709fcbecaecf100fd8398f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0437b0274a1da6956f2314e745a7af79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829580345\">\n<p id=\"fs-id1167836521713\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829628226\" class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of f of g of negative 3 to obtain that f of g of negative 3 equals f evaluated at g of negative 3. In the second step, we find g of negative 3 where g of x equals 3 x plus 2. This means that f of g of negative 3 equals f evaluated at 3 times negative 3 plus 2. In the third step, we simplify to obtain that f of g of negative 3 equals f of negative 7. In the fourth step, we find f of negative 7 where f of x equals x squared minus 4. This means that f of g of negative 3 equals negative 7 squared minus 4. In the fifth step, we simplify to obtain that f of g of negative 3 equals 45.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the definition of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-821d8aa5b85a9a4c8c2cfc072b13fa5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836698636\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833009341\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829807742\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of g of f of negative 1 to obtain that g of f of negative 1 equals g evaluated at f of negative 1. In the second step, we find f of negative 1 where f of x equals x squared minus 4. This means that g of f of negative 1 equals g of negative 1 squared minus 4. In the third step, we simplify. This means that g of f of negative 1 equals g of negative 3. In the fourth step, we find g of negative 3 where g of x equals 3 x plus 2. This means that g of f of negative 1 equals 3 times negative 3 plus 2. In the fifth step, we simplify. This means that g of f of negative 1 equals negative 7.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the definition of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f41f508742b0663a1da34ef42c0d7de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836623006\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829589897\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830123199\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829634197\" class=\"unnumbered unstyled\" summary=\"In the first step, we use the definition of f of f of 2 to obtain that f of f of 2 equals f evaluated at f of 2. In the second step, we find f of x where f of x equals x squared minus 4. This means that f of f of 2 equals f of 2 squared minus 4. In the third step, we simplify to obtain that f of f of 2 equals f of 0. In the fourth step, we find f of 0 where f of x equals x squared minus 4. This means that f of f of 2 equals 0 squared minus 4. In the fifth step, we simplify to obtain that f of f of 2 equals negative 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the definition of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0437b0274a1da6956f2314e745a7af79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836319358\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836730743\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836325800\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836620982\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829579755\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836418899\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833087107\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_007g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836319449\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829744245\">\n<div data-type=\"problem\" id=\"fs-id1167829908684\">\n<p>For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1dab9fe2d9a3e5a61b5cd505fb70b59c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f43c3196cf798b23b64f9a1addda888_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#43;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d0e10dcc2ca4f87d90b8def1530fff3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3cbd29d47e0c8f9218ff1b9b89fc713c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0f52171f6b6a74e0f57942406755f97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><span class=\"token\">\u24d0<\/span> \u20138 <span class=\"token\">\u24d1<\/span> 5 <span class=\"token\">\u24d2<\/span> 40<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836561320\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833051710\">\n<div data-type=\"problem\" id=\"fs-id1167829579215\">\n<p id=\"fs-id1167829692973\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da7b0111ddcbe97d5bc9274041b6fc5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4bf3a51a33bc35a5fd1f325096a3ebbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#45;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b12fd5985cb4ff3f22237c1e1e0ff69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eae456e7707da489e04e66c6646c036c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12cc7acba2ca5af28c9146da402a47de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836688612\">\n<p id=\"fs-id1167829595371\"><span class=\"token\">\u24d0<\/span> 65 <span class=\"token\">\u24d1<\/span> 10 <span class=\"token\">\u24d2<\/span> 5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Determine Whether a Function is One-to-One<\/h3>\n<p id=\"fs-id1167830093543\">When we first introduced functions, we said a <span data-type=\"term\" class=\"no-emphasis\">function<\/span> is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each <em data-effect=\"italics\">x<\/em>-value is matched with only one <em data-effect=\"italics\">y<\/em>-value.<\/p>\n<p id=\"fs-id1167836340795\">We used the birthday example to help us understand the definition. Every person has a birthday, but no one has two birthdays and it is okay for two people to share a birthday. Since each person has exactly one birthday, that relation is a function.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829586481\" data-alt=\"This figure shows two tables. To the left is the table labeled Name, which from top to bottom reads Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, and Danny. The table on the right is labeled Birthday, which from top to bottom reads January 12, February 3, April 25, May 10, May 23, July 24, August 2, and September 15. There are arrows going from Alison to April 25, Penelope to May 23, June to August 2, Gregory to September 15, Geoffrey to January 12, Lauren to May 10, Stephen to July 24, Alice to February 3, Liz to July 24, and Danny to no birthday.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_008_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two tables. To the left is the table labeled Name, which from top to bottom reads Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, and Danny. The table on the right is labeled Birthday, which from top to bottom reads January 12, February 3, April 25, May 10, May 23, July 24, August 2, and September 15. There are arrows going from Alison to April 25, Penelope to May 23, June to August 2, Gregory to September 15, Geoffrey to January 12, Lauren to May 10, Stephen to July 24, Alice to February 3, Liz to July 24, and Danny to no birthday.\" \/><\/span><\/p>\n<p id=\"fs-id1167829789822\">A function is <span data-type=\"term\">one-to-one<\/span> if each value in the range has exactly one element in the domain. For each ordered pair in the function, each <em data-effect=\"italics\">y<\/em>-value is matched with only one <em data-effect=\"italics\">x<\/em>-value.<\/p>\n<p id=\"fs-id1167836516310\">Our example of the birthday relation is not a one-to-one function. Two people can share the same birthday. The range value August 2 is the birthday of Liz and June, and so one range value has two domain values. Therefore, the function is not one-to-one.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836543477\">\n<div data-type=\"title\">One-to-One Function<\/div>\n<p>A function is <strong data-effect=\"bold\">one-to-one<\/strong> if each value in the range corresponds to one element in the domain. For each ordered pair in the function, each <em data-effect=\"italics\">y<\/em>-value is matched with only one <em data-effect=\"italics\">x<\/em>-value. There are no repeated <em data-effect=\"italics\">y<\/em>-values.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836548952\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829589807\">\n<p>For each set of ordered pairs, determine if it represents a function and, if so, if the function is one-to-one.<\/p>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ddf71187d01ae10ad8bd117425250c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\" \/> and <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ddf682f0dbea8455aeff740a9e8575f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"273\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836620263\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6da20cc58caced13ea1b6fd22fa65906_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\" \/><\/p>\n<p>Each <em data-effect=\"italics\">x<\/em>-value is matched with only one <em data-effect=\"italics\">y<\/em>-value. So this relation is a function.<\/p>\n<p id=\"fs-id1167836684567\">But each <em data-effect=\"italics\">y<\/em>-value is not paired with only one <em data-effect=\"italics\">x<\/em>-value, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-861b0c9f5d9dcab7db8f9e369b80649a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4842b114076573d68a06191720ac1828_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> for example. So this function is not one-to-one.<\/p>\n<p id=\"fs-id1167836531055\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fcfed16a05aa63a1fef3c15b096bce6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"265\" style=\"vertical-align: -5px;\" \/><\/p>\n<p>Each <em data-effect=\"italics\">x<\/em>-value is matched with only one <em data-effect=\"italics\">y<\/em>-value. So this relation is a function.<\/p>\n<p id=\"fs-id1167836546333\">Since each <em data-effect=\"italics\">y<\/em>-value is paired with only one <em data-effect=\"italics\">x<\/em>-value, this function is one-to-one.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829627366\">\n<div data-type=\"problem\" id=\"fs-id1167829695780\">\n<p id=\"fs-id1167836730111\">For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.<\/p>\n<p id=\"fs-id1167836689409\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f48e5ec2adae6151a69d1fd21e4ad40c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"439\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9fd54197074d4ca3f2cae4d372e484f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"398\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836553296\">\n<p id=\"fs-id1167836558408\"><span class=\"token\">\u24d0<\/span> One-to-one function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Function; not one-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836423521\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836389505\">\n<div data-type=\"problem\" id=\"fs-id1167829753177\">\n<p>For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.<\/p>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01651b3066118186fb68d7d296db6b93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#55;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#55;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96a39cf77de697f3337c6b21d3ffeb38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"439\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836628719\">\n<p><span class=\"token\">\u24d0<\/span> Not a function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Function; not one-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836737936\">To help us determine whether a relation is a function, we use the <span data-type=\"term\" class=\"no-emphasis\">vertical line test<\/span>. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. Also, if any vertical line intersects the graph in more than one point, the graph does not represent a function.<\/p>\n<p id=\"fs-id1167836697122\">The vertical line is representing an <em data-effect=\"italics\">x<\/em>-value and we check that it intersects the graph in only one <em data-effect=\"italics\">y<\/em>-value. Then it is a function.<\/p>\n<p id=\"fs-id1167833369216\">To check if a function is one-to-one, we use a similar process. We use a horizontal line and check that each horizontal line intersects the graph in only one point. The horizontal line is representing a <em data-effect=\"italics\">y<\/em>-value and we check that it intersects the graph in only one <em data-effect=\"italics\">x<\/em>-value. If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function. This is the <span data-type=\"term\">horizontal line test<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829712822\">\n<div data-type=\"title\">Horizontal Line Test<\/div>\n<p id=\"fs-id1167836416189\">If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function.<\/p>\n<\/div>\n<p id=\"fs-id1168757697826\">We can test whether a graph of a relation is a function by using the vertical line test. We can then tell if the function is one-to-one by applying the horizontal line test.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833257674\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829720214\">\n<div data-type=\"problem\" id=\"fs-id1167836692974\">\n<p id=\"fs-id1167836621109\">Determine <span class=\"token\">\u24d0<\/span> whether each graph is the graph of a function and, if so, <span class=\"token\">\u24d1<\/span> whether it is one-to-one.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836514026\" data-alt=\"This first graph shows a straight line passing through (0, 2) and (3, 0). This second shows a parabola opening up with vertex at (0, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_009_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This first graph shows a straight line passing through (0, 2) and (3, 0). This second shows a parabola opening up with vertex at (0, negative 1).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836685332\">\n<p id=\"fs-id1167836536170\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829693847\" data-alt=\"This figure shows a straight line passing through (0, 2) and (3, 0), with a red vertical line that only passes through one point and a blue horizontal line that only passes through one point.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_010_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line passing through (0, 2) and (3, 0), with a red vertical line that only passes through one point and a blue horizontal line that only passes through one point.\" \/><\/span><\/p>\n<p id=\"fs-id1167833224372\">Since any vertical line intersects the graph in at most one point, the graph is the graph of a function. Since any horizontal line intersects the graph in at most one point, the graph is the graph of a one-to-one function.<\/p>\n<p id=\"fs-id1167824739558\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829597115\" data-alt=\"This figure shows a parabola opening up with vertex at (0, negative 1), with a red vertical line that only passes through one point and a blue horizontal line that passes through two points.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_011_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening up with vertex at (0, negative 1), with a red vertical line that only passes through one point and a blue horizontal line that passes through two points.\" \/><\/span><\/p>\n<p id=\"fs-id1167836684122\">Since any vertical line intersects the graph in at most one point, the graph is the graph of a function. The horizontal line shown on the graph intersects it in two points. This graph does not represent a one-to-one function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836610440\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836729138\">\n<div data-type=\"problem\" id=\"fs-id1167836529356\">\n<p id=\"fs-id1167836732672\">Determine <span class=\"token\">\u24d0<\/span> whether each graph is the graph of a function and, if so, <span class=\"token\">\u24d1<\/span> whether it is one-to-one.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829688608\" data-alt=\"Graph a shows a parabola opening to the right with vertex at (negative 1, 0). Graph b shows an exponential function that does not cross the x axis and that passes through (0, 1) before increasing rapidly.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Graph a shows a parabola opening to the right with vertex at (negative 1, 0). Graph b shows an exponential function that does not cross the x axis and that passes through (0, 1) before increasing rapidly.\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829833822\">\n<p id=\"fs-id1167836630189\"><span class=\"token\">\u24d0<\/span> Not a function <span class=\"token\">\u24d1<\/span> One-to-one function<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833327286\">\n<div data-type=\"problem\" id=\"fs-id1167829893597\">\n<p id=\"fs-id1167829787025\">Determine <span class=\"token\">\u24d0<\/span> whether each graph is the graph of a function and, if so, <span class=\"token\">\u24d1<\/span> whether it is one-to-one.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836622399\" data-alt=\"Graph a shows a parabola opening up with vertex at (0, 3). Graph b shows a straight line passing through (0, negative 2) and (2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_013_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Graph a shows a parabola opening up with vertex at (0, 3). Graph b shows a straight line passing through (0, negative 2) and (2, 0).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829599594\">\n<p id=\"fs-id1167829809865\"><span class=\"token\">\u24d0<\/span> Function; not one-to-one <span class=\"token\">\u24d1<\/span> One-to-one function<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836509386\">\n<h3 data-type=\"title\">Find the Inverse of a Function<\/h3>\n<p id=\"fs-id1167829784507\">Let\u2019s look at a one-to one function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/>, represented by the ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b89fc4fab106642390f6827bd0f3795_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"214\" style=\"vertical-align: -5px;\" \/> For each <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-value, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> adds 5 to get the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-value. To \u2018undo\u2019 the addition of 5, we subtract 5 from each <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-value and get back to the original <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-value. We can call this \u201ctaking the inverse of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/>\u201d and name the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f86bde848b55015f22c61f89f878d41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"33\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836330596\" data-alt=\"This figure shows the set (0, 5), (1, 6), (2, 7) and (3, 8) on the left side of an oval. The oval contains the numbers 0, 1, 2, and 3. There are black arrows from these numbers that point to the numbers 5, 6, 7, and 8, respectively in a second oval to the right of the first. Above this, there is a black arrow labeled \u201cf add 5\u201d coming from the left oval to the right oval. There are red arrows from the numbers 5, 6, 7, and 8 in the right oval to the numbers 0, 1, 2, and 3, respectively, in the left oval. Below this, we have a red arrow labeled \u201cf with a superscript negative 1\u201d and \u201csubtract 5\u201d. To the right of this, we have the set (5, 0), (6, 1), (7, 2) and (8, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_014_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the set (0, 5), (1, 6), (2, 7) and (3, 8) on the left side of an oval. The oval contains the numbers 0, 1, 2, and 3. There are black arrows from these numbers that point to the numbers 5, 6, 7, and 8, respectively in a second oval to the right of the first. Above this, there is a black arrow labeled \u201cf add 5\u201d coming from the left oval to the right oval. There are red arrows from the numbers 5, 6, 7, and 8 in the right oval to the numbers 0, 1, 2, and 3, respectively, in the left oval. Below this, we have a red arrow labeled \u201cf with a superscript negative 1\u201d and \u201csubtract 5\u201d. To the right of this, we have the set (5, 0), (6, 1), (7, 2) and (8, 3).\" \/><\/span><\/p>\n<p id=\"fs-id1167836619718\">Notice that that the ordered pairs of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb35b54bbb6cdc3a372152553665e8b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -4px;\" \/> have their <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-values and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-values reversed. The domain of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> is the range of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb35b54bbb6cdc3a372152553665e8b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -4px;\" \/> and the domain of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb35b54bbb6cdc3a372152553665e8b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -4px;\" \/> is the range of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84385e96f3e4a161ac693910846e9e0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167836493064\">\n<div data-type=\"title\">Inverse of a Function Defined by Ordered Pairs<\/div>\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> is a one-to-one function whose ordered pairs are of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c92a92f915d1f75c2c7a9f50c608cedd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> then its inverse function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5563796ed237d9a9b52a4f18ebe9ff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/> is the set of ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fca552d0a0c5014bd6bdea824906b9f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#44;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<p id=\"fs-id1167836700044\">In the next example we will find the inverse of a function defined by ordered pairs.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836492441\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836537933\">\n<div data-type=\"problem\" id=\"fs-id1167829879570\">\n<p id=\"fs-id1167833007322\">Find the inverse of the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-327462921a2d6e8add80edc12dd71cea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"214\" style=\"vertical-align: -5px;\" \/> Determine the domain and range of the inverse function.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836328806\">This function is one-to-one since every <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-value is paired with exactly one <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-value.<\/p>\n<p id=\"fs-id1167836409343\">To find the inverse we reverse the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-values and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-values in the ordered pairs of the function.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-058b4070e8e83a3d25586751ff2963d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#118;&#101;&#114;&#115;&#101;&#32;&#70;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#111;&#109;&#97;&#105;&#110;&#32;&#111;&#102;&#32;&#73;&#110;&#118;&#101;&#114;&#115;&#101;&#32;&#70;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#51;&#44;&#53;&#44;&#55;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#97;&#110;&#103;&#101;&#32;&#111;&#102;&#32;&#73;&#110;&#118;&#101;&#114;&#115;&#101;&#32;&#70;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"503\" style=\"vertical-align: -38px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824736277\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824736280\">\n<div data-type=\"problem\" id=\"fs-id1167836531849\">\n<p id=\"fs-id1167836531851\">Find the inverse of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eeb05870e29866276e9ec1714032960b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"231\" style=\"vertical-align: -5px;\" \/> Determine the domain and range of the inverse function.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833135456\">\n<p id=\"fs-id1167833135458\">Inverse function: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-934aaa7c77ae733f9d0eb19f54a8b6a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"231\" style=\"vertical-align: -5px;\" \/> Domain: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b3a79fa6e94a7d8baf905302a900fe5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#52;&#44;&#55;&#44;&#49;&#48;&#44;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -5px;\" \/> Range: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-151e4a329980416b41a42737efa7eae6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1171792512793\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1171790741157\">\n<div data-type=\"problem\" id=\"fs-id1171790275560\">\n<p id=\"fs-id1171792545899\">Find the inverse of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f4ae4e71c67b2217724ef5a2018e965_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"269\" style=\"vertical-align: -5px;\" \/> Determine the domain and range of the inverse function.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171792843700\">\n<p id=\"fs-id1171790626716\">Inverse function: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a15f5e9f0ac7d81945d9cf98a604e026_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"269\" style=\"vertical-align: -5px;\" \/> Domain: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cbf105e4e0d4fb3f9a8701dd2ef9771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#48;&#44;&#49;&#44;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -5px;\" \/> Range: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b2e5a2ff02f5c91a5eea8a696383b0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#52;&#44;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833024456\">We just noted that if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> is a one-to-one function whose ordered pairs are of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c92a92f915d1f75c2c7a9f50c608cedd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> then its inverse function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5563796ed237d9a9b52a4f18ebe9ff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/> is the set of ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fca552d0a0c5014bd6bdea824906b9f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#44;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167832926088\">So if a point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7884c735dd07ca412df93caf3b3d0c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/> is on the graph of a function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccd462e56046be46a7c722c8d562e055_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/> then the ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80c3d0e5179b652c99fc9b74b6a4dc21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#44;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> is on the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-beeaa1c487e532ac3b081c190da76c09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"63\" style=\"vertical-align: -4px;\" \/> See <a href=\"#CNX_IntAlg_Figure_10_01_015\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_10_01_015\"><span data-type=\"media\" id=\"fs-id1167829593818\" data-alt=\"This figure shows the line y equals x with points (3,1) and (1,3) on either side of the line. These two points are connected by a dashed blue line segment.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_015.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the line y equals x with points (3,1) and (1,3) on either side of the line. These two points are connected by a dashed blue line segment.\" \/><\/span><\/div>\n<p id=\"fs-id1167829609063\">The distance between any two pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7884c735dd07ca412df93caf3b3d0c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80c3d0e5179b652c99fc9b74b6a4dc21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#44;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> is cut in half by the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7685b887e3d125a1e0ead8be22eccc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: -4px;\" \/> So we say the points are mirror images of each other through the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7685b887e3d125a1e0ead8be22eccc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167829746002\">Since every point on the graph of a function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> is a mirror image of a point on the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a8bac775e40f54abe4cfebdcbff812d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"63\" style=\"vertical-align: -4px;\" \/> we say the graphs are mirror images of each other through the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7685b887e3d125a1e0ead8be22eccc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: -4px;\" \/> We will use this concept to graph the inverse of a function in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829620960\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829620962\">\n<div data-type=\"problem\" id=\"fs-id1167829620964\">\n<p>Graph, on the same coordinate system, the inverse of the one-to one function shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829620970\" data-alt=\"This figure shows a line from (negative 5, negative 3) to (negative 3, negative 1) then to (negative 1,0) then to (0,2) and then to (3, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_016_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a line from (negative 5, negative 3) to (negative 3, negative 1) then to (negative 1,0) then to (0,2) and then to (3, 4).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836646061\">\n<p id=\"fs-id1167836646063\">We can use points on the graph to find points on the inverse graph. Some points on the graph are: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-088c797f16b38cba84c7338529ee329c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"307\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"fs-id1167836532837\">So, the inverse function will contain the points: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c33b691f52443e35c2439ef7d4385565_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"307\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832940223\" data-alt=\"This figure shows a line from (negative 5, negative 3) to (negative 3, negative 1) then to (negative 1, 0) then to (0,2) and then to (3, 4). Then there is a dashed line to denote y equals x. There is also a line from (negative 3, negative 5) to (negative 1, negative 3) then to (0, negative 1), then to (2, 0) and then to (4, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_017_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a line from (negative 5, negative 3) to (negative 3, negative 1) then to (negative 1, 0) then to (0,2) and then to (3, 4). Then there is a dashed line to denote y equals x. There is also a line from (negative 3, negative 5) to (negative 1, negative 3) then to (0, negative 1), then to (2, 0) and then to (4, 3).\" \/><\/span><\/p>\n<p id=\"fs-id1167832940234\">Notice how the graph of the original function and the graph of the inverse functions are mirror images through the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7685b887e3d125a1e0ead8be22eccc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826077493\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826077496\">\n<div data-type=\"problem\" id=\"fs-id1167826077498\">\n<p id=\"fs-id1167826077500\">Graph, on the same coordinate system, the inverse of the one-to one function.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167826077504\" data-alt=\"The graph shows a line from (negative 3, negative 4) to (negative 2, negative 2) then to (0, negative 1), then to (1, 2) and then to (4, 3). The graph shows a line from (negative 3, 4) to (0, 3) then to (1, 2) and then to (4, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_018_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows a line from (negative 3, negative 4) to (negative 2, negative 2) then to (0, negative 1), then to (1, 2) and then to (4, 3). The graph shows a line from (negative 3, 4) to (0, 3) then to (1, 2) and then to (4, 1).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826077515\"><span data-type=\"media\" id=\"fs-id1167826077518\" data-alt=\"This figure shows a line from (negative 4, negative 3) to (negative 2, negative 2) then to (negative 1, 0) then to (2, 1) and then to (3, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_301_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a line from (negative 4, negative 3) to (negative 2, negative 2) then to (negative 1, 0) then to (2, 1) and then to (3, 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836599086\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836599090\">\n<div data-type=\"problem\" id=\"fs-id1167836599092\">\n<p id=\"fs-id1167836599094\">Graph, on the same coordinate system, the inverse of the one-to one function.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1171790297071\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_022_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836599098\"><span data-type=\"media\" id=\"fs-id1167836599101\" data-alt=\"Graph extends from negative 4 to 4 on both axes. Points plotted are (negative 3, 4), (0, 3), (1, 2), and (4, 1). Line segments connect points.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Graph extends from negative 4 to 4 on both axes. Points plotted are (negative 3, 4), (0, 3), (1, 2), and (4, 1). Line segments connect points.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836625707\">When we began our discussion of an inverse function, we talked about how the inverse function \u2018undoes\u2019 what the original function did to a value in its domain in order to get back to the original <em data-effect=\"italics\">x<\/em>-value.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836625719\" data-alt=\"This figure shows x as the input to a box denoted as function f with f of x as the output of the box. Then, f of x is the input to a box denoted as function f superscript negative 1 with f superscript negative 1 of f of x equals x as the output of the box.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_019_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows x as the input to a box denoted as function f with f of x as the output of the box. Then, f of x is the input to a box denoted as function f superscript negative 1 with f superscript negative 1 of f of x equals x as the output of the box.\" \/><\/span><\/p>\n<div data-type=\"note\" id=\"fs-id1167836625730\">\n<div data-type=\"title\">Inverse Functions<\/div>\n<div data-type=\"equation\" id=\"fs-id1167836625735\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2c5e29f627677bb66bff77805ee480f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#32;&#97;&#108;&#108;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#100;&#111;&#109;&#97;&#105;&#110;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#32;&#97;&#108;&#108;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#100;&#111;&#109;&#97;&#105;&#110;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"376\" style=\"vertical-align: -18px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167829579603\">We can use this property to verify that two functions are inverses of each other.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829579606\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829579608\">\n<div data-type=\"problem\" id=\"fs-id1167829579610\">\n<p id=\"fs-id1167829579612\">Verify that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92e5b757359df95638b1b05a31301eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aab950119f36af41c95bb1190e09eb02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"88\" style=\"vertical-align: -6px;\" \/> are inverse functions.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836543935\">\n<p id=\"fs-id1167836543937\">The functions are inverses of each other if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5cbeff1e13407de2eb636db6b03a35f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72e124b9bf2c524fa53c15c28677b038_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<table id=\"fs-id1167829851289\" class=\"unnumbered unstyled\" summary=\"We want to examine whether g of f of x equals x In the first step, we substitute 5 x minus 1 for f of x which means that we are checking whether g of 5 x minus 1 equals x Then we find g of 5 x minus 1 where g of x equals the quantity x plus 1 all over 5. This means that we are trying to check whether the quantity 5 x plus 1 minus 1 all over 5 equals x. When we simplify we see that we are trying to check whether 5 x over 5 equals x. After simplifying further, we see that x equals x so this result holds. To investigate the other way, we are trying to determine whether f of g of x equals x. In the first step, we substitute the quantity x plus 1 all over 5 for g of x which means that we are trying to check whether f of quantity x plus 1 all over 5 equals x. Then we find f of quantity x plus 1 all over 5 where f of x equals 5 x minus 1. This means that we are trying to determine whether 5 times the quantity x plus 1 all over 5, minus 1 equals x. We simplify to see that to get x plus 1 minus 1 equals x. Simplifying further, we see that x does indeed equal x.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836524802\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-687571a11e62245057d8e5ce156fe0b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -1px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-066aa5868267297977626df4011032f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167825791236\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167825791245\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829715425\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829715444\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829696081\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9132b8a60ee6b0bd99e2957a96a54939_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"26\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9d4a03a388c1012a45f49710076105e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829718205\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829718228\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833202354\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832999569\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_020l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167832999579\">Since both <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5cbeff1e13407de2eb636db6b03a35f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4d5fafa9417f36ec74e373ef2ff9214_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/> are true, the functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92e5b757359df95638b1b05a31301eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aab950119f36af41c95bb1190e09eb02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"88\" style=\"vertical-align: -6px;\" \/> are inverse functions. That is, they are inverses of each other.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836399486\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836399489\">\n<div data-type=\"problem\" id=\"fs-id1167836399491\">\n<p id=\"fs-id1167836399493\">Verify that the functions are inverse functions.<\/p>\n<p id=\"fs-id1167836399496\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d68d2418825c52b01525faf11bd5f083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c522d00d69910bd5a2d39fe0392dbeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#51;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833020781\">\n<p id=\"fs-id1167833020783\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aabbb128e6046bd36c95a550443c867c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f15e32874e8f2786a5f5c6d9385c105b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> so they are inverses.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829715056\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829715060\">\n<div data-type=\"problem\" id=\"fs-id1167829715062\">\n<p id=\"fs-id1167829715064\">Verify that the functions are inverse functions.<\/p>\n<p id=\"fs-id1167829715067\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b302d32adafbd54f8084bfe3cece3142_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc03347661b30c34f1669ea669a00347_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#54;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829687013\">\n<p id=\"fs-id1167829687016\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aabbb128e6046bd36c95a550443c867c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f15e32874e8f2786a5f5c6d9385c105b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> so they are inverses.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836333437\">We have found inverses of function defined by ordered pairs and from a graph. We will now look at how to find an inverse using an algebraic equation. The method uses the idea that if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> is a one-to-one function with ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c92a92f915d1f75c2c7a9f50c608cedd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> then its inverse function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5563796ed237d9a9b52a4f18ebe9ff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/> is the set of ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fca552d0a0c5014bd6bdea824906b9f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#44;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167833345792\">If we reverse the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> in the function and then solve for <em data-effect=\"italics\">y<\/em>, we get our <span data-type=\"term\" class=\"no-emphasis\">inverse function<\/span>.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833345812\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Find the inverse of a One-to-One Function<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833239447\">\n<div data-type=\"problem\" id=\"fs-id1167833239449\">\n<p id=\"fs-id1167833239451\">Find the inverse of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-691e568e55ad49eff207ca68da31d9fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#43;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167821865222\"><span data-type=\"media\" id=\"fs-id1167821865224\" data-alt=\"Step 1 is to substitute y for f of x. To do so, we replace f of x with y. Hence, f of x equals 4 x plus 7 becomes y equals 4 x plus 7.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to substitute y for f of x. To do so, we replace f of x with y. Hence, f of x equals 4 x plus 7 becomes y equals 4 x plus 7.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167821865234\" data-alt=\"Step 2 is to interchange the variables x and y. To do so, we replace x with y and then y with x. Hence, we obtain x equals 4y plus 7.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to interchange the variables x and y. To do so, we replace x with y and then y with x. Hence, we obtain x equals 4y plus 7.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167821865243\" data-alt=\"Step 3 is to solve for y. To do so, we subtract 7 from each side and then divide by 4. Hence, we have x minus 7 equals 4y and then the quantity x minus 7 divided by 4 equals y.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve for y. To do so, we subtract 7 from each side and then divide by 4. Hence, we have x minus 7 equals 4y and then the quantity x minus 7 divided by 4 equals y.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836495426\" data-alt=\"Step 4 is to substitute f superscript negative 1 of x for y. To do so, we replace y with f superscript negative 1 of x. Hence, the quantity x minus 7 divided by 4 equals f superscript negative 1 of x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to substitute f superscript negative 1 of x for y. To do so, we replace y with f superscript negative 1 of x. Hence, the quantity x minus 7 divided by 4 equals f superscript negative 1 of x.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836495436\" data-alt=\"Step 5 is to verify that the functions are inverses. To do so, we show that f superscript negative 1 of f of x equals x and that f of f superscript negative 1of x equals x. Hence, we ask whether f inverse of 4x plus 7 equals x. This becomes a question of whether 4 x plus 7 minus 7 all divided by 4 equals x. This becomes a question of whether 4x divided by 4 equals x. This is true. To show the other side, we examine whether f of f inverse of x equals x. This becomes a question of whether f of the quantity x minus 7 divided by 4 equals x. This becomes a question of whether 4 times the quantity x minus 7 divided by 4 equals x. This becomes a question of whether x minus 7 plus 7 equals x. This is true.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_021e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to verify that the functions are inverses. To do so, we show that f superscript negative 1 of f of x equals x and that f of f superscript negative 1of x equals x. Hence, we ask whether f inverse of 4x plus 7 equals x. This becomes a question of whether 4 x plus 7 minus 7 all divided by 4 equals x. This becomes a question of whether 4x divided by 4 equals x. This is true. To show the other side, we examine whether f of f inverse of x equals x. This becomes a question of whether f of the quantity x minus 7 divided by 4 equals x. This becomes a question of whether 4 times the quantity x minus 7 divided by 4 equals x. This becomes a question of whether x minus 7 plus 7 equals x. This is true.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836495454\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836495457\">\n<div data-type=\"problem\" id=\"fs-id1167833239525\">\n<p id=\"fs-id1167833239527\">Find the inverse of the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55950c739f322be1dce43d92db3d3e59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#120;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833239556\">\n<p id=\"fs-id1167833239558\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39fb8661ec36a6cf5e7280b3da61938f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829909640\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829909643\">\n<div data-type=\"problem\" id=\"fs-id1167829909645\">\n<p id=\"fs-id1167829909647\">Find the inverse of the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e4b4cf12635a88a920098032977ac3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#120;&#43;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836537615\">\n<p id=\"fs-id1167836537617\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7857195e7c4bfc5ec1752cc9a67ea80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829738768\">We summarize the steps below.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829738771\" class=\"howto\">\n<div data-type=\"title\">How to Find the inverse of a One-to-One Function<\/div>\n<ol id=\"fs-id1167829738778\" class=\"stepwise\" type=\"1\">\n<li>Substitute <em data-effect=\"italics\">y<\/em> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-066aa5868267297977626df4011032f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Interchange the variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/li>\n<li>Solve for <em data-effect=\"italics\">y<\/em>.<\/li>\n<li>Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5563796ed237d9a9b52a4f18ebe9ff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/> for <em data-effect=\"italics\">y<\/em>.<\/li>\n<li>Verify that the functions are inverses.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167829650658\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Find the Inverse of a One-to-One Function<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829650664\">\n<div data-type=\"problem\" id=\"fs-id1167833350434\">\n<p id=\"fs-id1167833350436\">Find the inverse of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80221a22e1ca38c3f87106943fbd54f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#120;&#45;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826172026\">\n<p id=\"fs-id1167826172028\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e99c1ef1c1e0620e70b5ffce2f9213e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#120;&#45;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#120;&#45;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#116;&#101;&#114;&#99;&#104;&#97;&#110;&#103;&#101;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#121;&#45;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#121;&#45;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#121;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#43;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#43;&#51;&#125;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#43;&#51;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"183\" width=\"529\" style=\"vertical-align: -87px;\" \/><\/p>\n<p id=\"fs-id1167836399012\">Verify that the functions are inverses.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-750bf44cad5005266b8778118a0a8514_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#120;&#45;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#43;&#51;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#120;&#45;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#125;&#43;&#51;&#125;&#123;&#50;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#43;&#51;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#45;&#51;&#43;&#51;&#125;&#123;&#50;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#43;&#51;&#45;&#51;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#125;&#123;&#50;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836480963\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836480966\">\n<div data-type=\"problem\" id=\"fs-id1167836480968\">\n<p id=\"fs-id1167836480970\">Find the inverse of the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8614976858a7de7634efae793cafd030_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#51;&#120;&#45;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829832059\">\n<p id=\"fs-id1167829832062\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3c3b1fb059dfd8acb54d18b846179b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#43;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"114\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833020910\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833020913\">\n<div data-type=\"problem\" id=\"fs-id1167833020915\">\n<p id=\"fs-id1167833020917\">Find the inverse of the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51e0ad1907fb4285d41ec6b2d5924c0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#54;&#120;&#45;&#55;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836477385\">\n<p id=\"fs-id1167836477387\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ee3ad74023f3600ccace4be55910624_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#55;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"114\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829906171\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167829906177\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Composition of Functions:<\/strong> The composition of functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa307d44fe899099cad9fc84395f6eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/> is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ca4cf2da3c03e0d5a112084842d8569_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"36\" style=\"vertical-align: -4px;\" \/> and is defined by\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167829879433\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93ad33b10518db26515730aecb3748be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"162\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"newline\"><\/div>\n<p>We read <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-126f21dc35931f9b8d72c325f659566b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Horizontal Line Test:<\/strong> If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.<\/li>\n<li><strong data-effect=\"bold\">Inverse of a Function Defined by Ordered Pairs:<\/strong> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> is a one-to-one function whose ordered pairs are of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c92a92f915d1f75c2c7a9f50c608cedd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> then its inverse function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5563796ed237d9a9b52a4f18ebe9ff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/> is the set of ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fca552d0a0c5014bd6bdea824906b9f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#44;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Inverse Functions:<\/strong> For every <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> in the domain of one-to-one function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9d003671d422fc30e48900ad5c5855d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"33\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167836544056\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce7029f52dba438d5ce54fabd72f6393_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"142\" style=\"vertical-align: -18px;\" \/><\/div>\n<\/li>\n<li><strong data-effect=\"bold\">How to Find the Inverse of a One-to-One Function:<\/strong>\n<ol id=\"fs-id1167836768376\" class=\"stepwise\" type=\"1\">\n<li>Substitute <em data-effect=\"italics\">y<\/em> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-066aa5868267297977626df4011032f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Interchange the variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/li>\n<li>Solve for <em data-effect=\"italics\">y<\/em>.<\/li>\n<li>Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5563796ed237d9a9b52a4f18ebe9ff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62f853fa6f372493298c507883a9f490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Verify that the functions are inverses.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833020689\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167833020692\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167833020699\"><strong data-effect=\"bold\">Find and Evaluate Composite Functions<\/strong><\/p>\n<p id=\"fs-id1167833020704\">In the following exercises, find <span class=\"token\">\u24d0<\/span> (<em data-effect=\"italics\">f<\/em> \u2218 <em data-effect=\"italics\">g<\/em>)(<em data-effect=\"italics\">x<\/em>), <span class=\"token\">\u24d1<\/span> (<em data-effect=\"italics\">g<\/em> \u2218 <em data-effect=\"italics\">f<\/em>)(<em data-effect=\"italics\">x<\/em>), and <span class=\"token\">\u24d2<\/span> (<em data-effect=\"italics\">f<\/em> \u00b7 <em data-effect=\"italics\">g<\/em>)(<em data-effect=\"italics\">x<\/em>).<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829721059\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829721063\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-613893b4d623a19873160a67cecf901e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-276faa7bab4090028bb72145cbea3a07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829650613\">\n<p id=\"fs-id1167829650615\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2dc48e4f56832169814f986e915cb338_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#120;&#43;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"59\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-517a2b73ae9fc5cce73c9b2fcdad8d33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#120;&#43;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"58\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b215ef9c90ef01eb25a3ff7c5ca0f11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#54;&#120;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"115\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825708408\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167825708410\">\n<p id=\"fs-id1167825708412\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d0823a4beca7aac4c7333e810fa3362_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0ede1072d296c92d65debaabf924951_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826025403\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826025405\">\n<p id=\"fs-id1167826025407\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18b6278bb544b83d107b5427347bcb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d408df7b72b0e51ac8d46178ee2335d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836790650\">\n<p id=\"fs-id1167836790652\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d387038cb383f32d8172951b05f4bda0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"58\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39bcee4be9e6af0b1317147158746292_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#120;&#45;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"68\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47cd771f536ffef428b3893765d1a149_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#57;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"115\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829878007\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829878009\">\n<p id=\"fs-id1167829878011\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-973c06d333a495d183199822f9aff216_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a75efe7e6979bc3115f409e747d1a578_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829696400\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829696402\">\n<p id=\"fs-id1167829696404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-435af59836f3286bcd17e175f2839b53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1a5f6e851379f47f60b52721550f469_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829830707\">\n<p id=\"fs-id1167829830709\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-974e91acd6a8e34052d66a4f6b052fd2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"67\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d018ab43456faf889406c7aef8b9219b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"75\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffaeb87f92ab67ba9c54e2b93dc12fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"74\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829840934\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829840936\">\n<p id=\"fs-id1167829840938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d268a37dadc23b636bc6955a3e233880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fdc487214399f8d8b8e31716350260c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836626117\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836626119\">\n<p id=\"fs-id1167836626121\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7eeadb998ce7a2ec9a62c2c28141215c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ded55672642c130f77e58b8af6918e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825872598\">\n<p id=\"fs-id1167825872600\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9943b96b392e3b04e3de1658074a415_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"57\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8ef8f95978d043b887472ed0954c3b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21f58e3dcdb2c09f5e42c3fcb705270b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167822996884\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167822996887\">\n<p id=\"fs-id1167822996889\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-613893b4d623a19873160a67cecf901e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d41b0645aab6311e7e4ca56cb76fa9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836602658\">In the following exercises, find the values described.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836602662\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836602666\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33350c8e048f2edc0e0aea1b05a96a4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"118\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b38329640547decd5a3ec727f3ebbda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#120;&#45;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f6f5048ba15da6aa380a341288665f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24d3504dcaae670dd7d1a5888ef59f32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c87f9196bf1af3ffcedd54427cf7efb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833138071\">\n<p id=\"fs-id1167833138073\"><span class=\"token\">\u24d0<\/span> 245 <span class=\"token\">\u24d1<\/span> 104 <span class=\"token\">\u24d2<\/span> 53<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836607003\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836607006\">\n<p id=\"fs-id1167836607008\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62cc62a365f56f4ece4c1b8025555248_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b578787c94e10aafe453692712023d3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#45;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3c41e2d5cb1970d61c95cbb6f8c6397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f67a2720907ee945f9f25df39144bad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4829011a77d88694da9f0beb39a9d52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"76\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829807251\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829807254\">\n<p id=\"fs-id1167829807256\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e307eb250dd2f2ee20d3b62982dcc411_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8a850ba7088da072f6a2a1d6c764da6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4d68abf9978522d83c24540ca0a2e9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06e4bb5082f008814047610c4cd6a0a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b5edb61600cd83a0bb74c437204f167_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829833766\">\n<p id=\"fs-id1167829833768\"><span class=\"token\">\u24d0<\/span> 250 <span class=\"token\">\u24d1<\/span> 14 <span class=\"token\">\u24d2<\/span> 77<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836690124\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836690126\">\n<p id=\"fs-id1167836690128\">For functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63189d2628c75f0bd6ae0f506961bd91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23b8d0e0f3ae768ef19aac54cc08d0b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f6f5048ba15da6aa380a341288665f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f67a2720907ee945f9f25df39144bad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#102;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b5edb61600cd83a0bb74c437204f167_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#99;&#105;&#114;&#99;&#32;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167825808240\"><strong data-effect=\"bold\">Determine Whether a Function is One-to-One<\/strong><\/p>\n<p id=\"fs-id1167825808246\">In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167825808250\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167825808252\">\n<p id=\"fs-id1167825808254\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3da7d797f119f53d286212c38ebd2b44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"239\" style=\"vertical-align: -5px;\" \/>,<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce1d3820b06b395c92135828705ec713_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836530984\">\n<p id=\"fs-id1167824668944\">Function; not one-to-one<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824668950\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824668952\">\n<p id=\"fs-id1167824668954\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e653d4691fdbd7e6d142d486b080f44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"239\" style=\"vertical-align: -5px;\" \/>,<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-621359f0c4d667de85b020cbb9c8b445_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833369697\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833369699\">\n<p id=\"fs-id1167833369701\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60fa88b21717f7efc6c61df6a6cb50e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"230\" style=\"vertical-align: -5px;\" \/>,<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c74eab2ec77a4f75cf644f445e7fc96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824704165\">\n<p id=\"fs-id1167824704167\">One-to-one function<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824704172\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824704174\">\n<p id=\"fs-id1167824704177\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76db51d6b08b6c12c7d16f356a30ff52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"197\" style=\"vertical-align: -5px;\" \/>,<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99710d90d4919dc09d58b1da471e549c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"193\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836533576\">In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836533580\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836533582\">\n<p id=\"fs-id1167836533584\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829787690\" data-alt=\"This figure shows a graph of a circle with center at the origin and radius 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a graph of a circle with center at the origin and radius 3.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829787704\" data-alt=\"This figure shows a graph of a parabola opening upward with vertex at (0k, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_202_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a graph of a parabola opening upward with vertex at (0k, 2).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829787714\">\n<p id=\"fs-id1167829787716\"><span class=\"token\">\u24d0<\/span> Not a function <span class=\"token\">\u24d1<\/span> Function; not one-to-one<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829787729\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829787731\">\n<p id=\"fs-id1167829871910\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829871917\" data-alt=\"This figure shows a parabola opening to the right with vertex at (negative 2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_203_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening to the right with vertex at (negative 2, 0).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829871932\" data-alt=\"This figure shows a graph of a polynomial with odd order, so that it starts in the third quadrant, increases to the origin and then continues increasing through the first quadrant.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_204_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a graph of a polynomial with odd order, so that it starts in the third quadrant, increases to the origin and then continues increasing through the first quadrant.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167823012125\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167823012127\">\n<p id=\"fs-id1167823012129\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167823012136\" data-alt=\"This figure shows a graph of a curve that starts at (negative 6 negative 2) increases to the origin and then continues increasing slowly to (6, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_205_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a graph of a curve that starts at (negative 6 negative 2) increases to the origin and then continues increasing slowly to (6, 2).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167823012152\" data-alt=\"This figure shows a parabola opening upward with vertex at (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_206_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening upward with vertex at (0, negative 4).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167823012162\">\n<p id=\"fs-id1167823012164\"><span class=\"token\">\u24d0<\/span> One-to-one function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Function; not one-to-one<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829701984\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829701986\">\n<p id=\"fs-id1167829701988\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829701994\" data-alt=\"This figure shows a straight line segment decreasing from (negative 4, 6) to (2, 0), after which it increases from (2, 0) to (6, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_207_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line segment decreasing from (negative 4, 6) to (2, 0), after which it increases from (2, 0) to (6, 4).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829702010\" data-alt=\"This figure shows a circle with radius 4 and center at the origin.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_208_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a circle with radius 4 and center at the origin.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836624981\">In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836624985\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836624987\">\n<p id=\"fs-id1167836624990\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4740ae7ec93f91f7379265f9a2a0a56c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"206\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829808255\">\n<p id=\"fs-id1167829808257\">Inverse function: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8c9e8b6b54aa83fcf61e88b77092126_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"214\" style=\"vertical-align: -5px;\" \/> Domain: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1ef958c8aa62c952393a8caab0eb39e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#49;&#44;&#50;&#44;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -5px;\" \/> Range: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17ccfd75f2b83976203d461e8dcffc4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#50;&#44;&#52;&#44;&#54;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836664617\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836664619\">\n<p id=\"fs-id1167829732032\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e46112e833fd14e9fec5bb5265b2216_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#44;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"233\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825703072\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167825703074\">\n<p id=\"fs-id1167825703077\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9fed284b5eb764517485e93751c0b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"229\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829890422\">\n<p id=\"fs-id1167829890424\">Inverse function: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f59e9710dfed1bd6e14092167b6bc37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"236\" style=\"vertical-align: -5px;\" \/> Domain: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e64fcc1bf430875610e6121b9aa5269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#50;&#44;&#51;&#44;&#55;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" style=\"vertical-align: -5px;\" \/> Range: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-151e4a329980416b41a42737efa7eae6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829609019\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829609022\">\n<p id=\"fs-id1167829609024\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6e7605a876350e4d8baf88148bea9f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"206\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830019130\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829807304\">\n<p id=\"fs-id1167829807306\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0682a42ecad5aa4e3a031def124869b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"261\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832982002\">\n<p id=\"fs-id1167832982004\">Inverse function: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-497a39ba2a272dfd9927f5c274ada773_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"255\" style=\"vertical-align: -5px;\" \/> Domain: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c170e68889b7bf3294f20dc2cd0c392_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#44;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -5px;\" \/> Range: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6aa2c864a50530e68c2c06e05fc7836a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829957984\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829957986\">\n<p id=\"fs-id1167829957988\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-250a74c93499b46a479e5b66fdb98e90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"206\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829690781\">In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829690786\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829690788\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829690791\" data-alt=\"This figure shows a series of line segments from (negative 4, negative 3) to (negative 3, 0) then to (negative 1, 2) and then to (3, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_209_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 4, negative 3) to (negative 3, 0) then to (negative 1, 2) and then to (3, 4).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829690802\"><span data-type=\"media\" id=\"fs-id1167829690805\" data-alt=\"This figure shows a series of line segments from (negative 3, negative 4) to (0, negative 3) then to (2, negative 1), and then to (4, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 3, negative 4) to (0, negative 3) then to (2, negative 1), and then to (4, 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824732135\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824732137\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167824732140\" data-alt=\"This figure shows a series of line segments from (negative 4, negative 4) to (negative 3, 1) then to (0, 2) and then to (2, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_210_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 4, negative 4) to (negative 3, 1) then to (0, 2) and then to (2, 4).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824732166\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824732168\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167824732171\" data-alt=\"This figure shows a series of line segments from (negative 4, 4) to (0, 3) then to (3, 2) and then to (4, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_211_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 4, 4) to (0, 3) then to (3, 2) and then to (4, negative 1).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829840824\"><span data-type=\"media\" id=\"fs-id1167829840828\" data-alt=\"This figure shows a series of line segments from (negative 1, 4) to (2, 3) then to (3, 0), and then to (4, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 1, 4) to (2, 3) then to (3, 0), and then to (4, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829840839\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829840841\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829840845\" data-alt=\"This figure shows a series of line segments from (negative 4, negative 4) to (negative 1, negative 3) then to (0, 1), then to (1, 3), and then to (4, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_212_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a series of line segments from (negative 4, negative 4) to (negative 1, negative 3) then to (0, 1), then to (1, 3), and then to (4, 4).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836523852\">In the following exercises, determine whether or not the given functions are inverses.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836523856\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836523858\">\n<p id=\"fs-id1167836523860\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32529ec0b73f55b4fc86771c100fdeeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c62d921aa863e831d462d57c51cc156e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824733031\">\n<p id=\"fs-id1167824733033\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aabbb128e6046bd36c95a550443c867c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f15e32874e8f2786a5f5c6d9385c105b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> so they are inverses.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832936881\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832936883\">\n<p id=\"fs-id1167832936885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7e20fd56f97b7ae91d6efd359672d46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-356d6aefd89763cad41d984c36832dce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832926058\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832926061\">\n<p id=\"fs-id1167832926063\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4665c5070f05b6da15c378e6345d280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c657e71603b8b474f058044967d7b9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829787270\">\n<p id=\"fs-id1167829787272\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aabbb128e6046bd36c95a550443c867c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f15e32874e8f2786a5f5c6d9385c105b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> so they are inverses.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825003602\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167825003604\">\n<p id=\"fs-id1167825003606\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8246f3f49f8ae05f5898a0fb3546ac86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"77\" style=\"vertical-align: -7px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-104d1da5fdf238814abfd4307410af23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#49;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829808958\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829808960\">\n<p id=\"fs-id1167829808962\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3ce0ae5404c1abf5187b612471592f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00d82de3d1cfb141bd272ff0c6afe7c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#51;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836440128\">\n<p id=\"fs-id1167836440130\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aabbb128e6046bd36c95a550443c867c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f15e32874e8f2786a5f5c6d9385c105b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> so they are inverses.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824726024\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829832983\">\n<p id=\"fs-id1167829832985\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc7d126b46f521cb41a2673c446d58d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6cafc5a1e705f231b2a3777636ee1cce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824765407\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824765409\">\n<p id=\"fs-id1167824765411\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa15cd2c507cdbf4bd9fed5af50d861f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55d55c4636f5d7c5b85dcf8f4ce07d31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833051870\">\n<p id=\"fs-id1167833051872\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aabbb128e6046bd36c95a550443c867c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f15e32874e8f2786a5f5c6d9385c105b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> so they are inverses (for nonnegative <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c74d4b67ba779d1bd8a6bf74f9f0e50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829849326\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829849328\">\n<p id=\"fs-id1167829849330\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a0b1d4f74299476cac3e87aff23124d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"118\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e282d942cedd4a0c6514f4082df49355_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833022953\">In the following exercises, find the inverse of each function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833022957\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833022959\">\n<p id=\"fs-id1167833022961\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0935147817c6373ce042e7eea52d174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833022984\">\n<p id=\"fs-id1167833022986\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-491ef42cf8dc60a11b6233d8c32057fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8abdaf6dd9677f60c8bdf7858ce6403_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#120;&#43;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829859344\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829859346\">\n<p id=\"fs-id1167829859348\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7005353689e04dab7a7c6322a9ddb2a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824602071\">\n<p id=\"fs-id1167824602073\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c0524b20361972a8f7d6e3af7af7bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824602102\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824602104\">\n<p id=\"fs-id1167824602106\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57c47ca219e40435aafcfcfd46d20ae3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829784467\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829784470\">\n<p id=\"fs-id1167829784472\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-974517e33680a3bd3a7f0cb6619a3eba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"71\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829620578\">\n<p id=\"fs-id1167829620580\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ff65809a26c458584a64b4998b94539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829620608\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829620610\">\n<p id=\"fs-id1167829620612\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e36669d92770ea7ef574558cea762b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"71\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836621636\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836621638\">\n<p id=\"fs-id1167836621640\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2621340f9a6d8939f8c47089377e6a20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829811511\">\n<p id=\"fs-id1167829811513\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8240ec4bfd7fdcd2e19e8269a2c0fa83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#55;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829811547\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829811549\">\n<p id=\"fs-id1167829811552\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd1cbaf7578b981f865f3048e8853059_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829801847\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829801849\">\n<p id=\"fs-id1167829801851\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7751d144f04f4dcbbc1c2de64b9838e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833175196\">\n<p id=\"fs-id1167833175199\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-727b1aafbb771ee7bf57e4dbb288f0f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#53;&#125;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829715177\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829715179\">\n<p id=\"fs-id1167829715182\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5eb6226c4768a377a7a563a3386892b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#53;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825003647\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167825003650\">\n<p id=\"fs-id1167825003652\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b05094c17b4c927feece676b2a6183d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46bb299f47d94e1927057d14ab78801f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825003690\">\n<p id=\"fs-id1167825003692\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08929d1515ca86f2140e6b21fd4d78c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826122987\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826122990\">\n<p id=\"fs-id1167826122992\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1dab9fe2d9a3e5a61b5cd505fb70b59c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46bb299f47d94e1927057d14ab78801f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829851254\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829851256\">\n<p id=\"fs-id1167829851258\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51a62998b2ead4b212996c38e190108a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829651284\">\n<p id=\"fs-id1167829651286\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bed3df80042d28e18f8b8b3b27adffbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829651320\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829651322\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b97f0ea9dc89072dc8a303376ab3de48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836575720\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836575722\">\n<p id=\"fs-id1167836575724\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-792a544a63fd53c3d2535c28d363a170_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"89\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836575752\">\n<p id=\"fs-id1167836575754\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d30c2a1a5ac1ef44dbf80b190c678b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836495478\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836495480\">\n<p id=\"fs-id1167836495483\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3acc88e58e9f9ffb8b3fc9bc7d2783f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829704844\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829704847\">\n<p id=\"fs-id1167829704849\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9122eb6639d2e30fe1d399f32e7ccbd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05f93945a061277e0c2f50680ad2a535_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829704886\">\n<p id=\"fs-id1167829704888\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67ca82280787f0849b14b7eee7418d44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -4px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46bb299f47d94e1927057d14ab78801f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167821885714\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167821885716\">\n<p id=\"fs-id1167821885718\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42f39bd3a12c20a852ce6d91da76b0d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#56;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cf8f2f01b9876a29007c7e0a0a331a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836494143\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836494145\">\n<p id=\"fs-id1167836494147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7bb1bcfde5da03037c5d329eccc7fd87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836494175\">\n<p id=\"fs-id1167836494177\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30a1dfe46829166c46d341b6d0bbc67e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833084919\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833084921\">\n<p id=\"fs-id1167833084924\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d737debb1da797903a4ce5db03a8483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824917078\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824917080\">\n<p id=\"fs-id1167824917082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82ff0523d8f21766fb51a1c1f07b355b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#57;&#120;&#45;&#53;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b9194bc7dd72c0746871b02725b4614_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836450145\">\n<p id=\"fs-id1167836450147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa17a328e5f2a62c5b97e3ffe73e4ce9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#53;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"114\" style=\"vertical-align: -6px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46bb299f47d94e1927057d14ab78801f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836450194\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836450196\">\n<p id=\"fs-id1167836450198\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdb82c81eabe2c3bd426d95d087b4e88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#56;&#120;&#45;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aceceba5d34ec793eebd95473f1ad75d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829808436\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829808438\">\n<p id=\"fs-id1167829808440\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7bc8aca2eee18b727008592a593af182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#45;&#51;&#120;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829808470\">\n<p id=\"fs-id1167829808472\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83c75b1c2fc0f48a4e23fb87dc7c81cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#45;&#53;&#125;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"114\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836480837\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836480839\">\n<p id=\"fs-id1167836480841\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e256a176ae397c94c6c9575bfd5404ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#45;&#52;&#120;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836514906\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167836514914\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836514916\">\n<p id=\"fs-id1167836514918\">Explain how the graph of the inverse of a function is related to the graph of the function.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836514923\">\n<p id=\"fs-id1167829751136\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829751141\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829751144\">\n<p id=\"fs-id1167829751146\">Explain how to find the inverse of a function from its equation. Use an example to demonstrate the steps.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167829751165\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829751177\" data-alt=\"This table has four rows and four columns. The first row, which serves as a header, reads I can\u2026, Confidently, With some help, and No\u2014I don\u2019t get it. The first column below the header row reads Find and evaluate composite functions, determine whether a function is one-to-one, and find the inverse of a function. The rest of the cells are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_10_01_213_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four rows and four columns. The first row, which serves as a header, reads I can\u2026, Confidently, With some help, and No\u2014I don\u2019t get it. The first column below the header row reads Find and evaluate composite functions, determine whether a function is one-to-one, and find the inverse of a function. The rest of the cells are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167829751188\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p>\n<p id=\"fs-id1167829751196\">\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p>\n<p id=\"fs-id1167836399239\">\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p>\n<p id=\"fs-id1167836399250\">\u2026no\u2014I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl>\n<dt>one-to-one function<\/dt>\n<dd id=\"fs-id1167836399269\">A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each <em data-effect=\"italics\">y<\/em>-value is matched with only one <em data-effect=\"italics\">x<\/em>-value.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":146,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4255","chapter","type-chapter","status-publish","hentry"],"part":14505,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4255","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4255\/revisions"}],"predecessor-version":[{"id":14508,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4255\/revisions\/14508"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/14505"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4255\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=4255"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=4255"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=4255"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=4255"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}