{"id":2017,"date":"2018-12-11T13:35:37","date_gmt":"2018-12-11T18:35:37","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/relations-and-functions\/"},"modified":"2018-12-11T13:35:37","modified_gmt":"2018-12-11T18:35:37","slug":"relations-and-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/relations-and-functions\/","title":{"raw":"Relations and Functions","rendered":"Relations and Functions"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Find the domain and range of a relation<\/li><li>Determine if a relation is a function<\/li><li>Find the value of a function<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167836299681\" class=\"be-prepared\"><p id=\"fs-id1167829627915\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167836635622\" type=\"1\"><li>Evaluate \\(3x-5\\) when \\(x=-2\\).<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Evaluate \\(2{x}^{2}-x-3\\) when \\(x=a.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(7x-1-4x+5.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836652573\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829789538\"><h3 data-type=\"title\">Find the Domain and Range of a Relation<\/h3><p id=\"fs-id1167836538118\">As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number and our birthday are matched to our name. There is a relationship between our name and each of those items.<\/p><p id=\"fs-id1167829921677\">When your professor gets her class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student\u2019s ID number, we call this a <span data-type=\"term\">relation<\/span>.<\/p><div data-type=\"equation\" id=\"fs-id1167836418492\" class=\"unnumbered\" data-label=\"\">\\(\\text{(Student name, Student ID #)}\\)<\/div><p id=\"fs-id1167836547061\">The set of all the names of the students in the class is called the <span data-type=\"term\">domain<\/span> of the relation and the set of all student ID numbers paired with these students is the range of the relation.<\/p><p id=\"fs-id1167833059636\">There are many similar situations where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation.<\/p><div data-type=\"note\" id=\"fs-id1167836378970\"><div data-type=\"title\">Relation<\/div><p id=\"fs-id1167824735594\">A <strong data-effect=\"bold\">relation<\/strong> is any set of ordered pairs,\\(\\left(x,y\\right).\\) All the <em data-effect=\"italics\">x<\/em>-values in the ordered pairs together make up the <strong data-effect=\"bold\">domain<\/strong>. All the <em data-effect=\"italics\">y<\/em>-values in the ordered pairs together make up the <strong data-effect=\"bold\">range<\/strong>.<\/p><\/div><div data-type=\"example\" id=\"fs-id1167836692527\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836327016\"><div data-type=\"problem\" id=\"fs-id1167833385954\"><p id=\"fs-id1167826194732\">For the relation \\(\\left\\{\\left(1,1\\right),\\left(2,4\\right),\\left(3,9\\right),\\left(4,16\\right),\\left(5,25\\right)\\right\\}:\\)<\/p><p id=\"fs-id1167836619755\"><span class=\"token\">\u24d0<\/span> Find the domain of the relation.<\/p><p id=\"fs-id1167833356505\"><span class=\"token\">\u24d1<\/span> Find the range of the relation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824765140\"><p id=\"fs-id1167836597346\">\\(\\phantom{\\rule{21em}{0ex}}\\left\\{\\left(1,1\\right),\\left(2,4\\right),\\left(3,9\\right),\\left(4,16\\right),\\left(5,25\\right)\\right\\}\\)<\/p><p id=\"fs-id1167829691118\"><span class=\"token\">\u24d0<\/span> The domain is the set of all <em data-effect=\"italics\">x<\/em>-values of the relation. \\(\\phantom{\\rule{3.3em}{0ex}}\\left\\{1,2,3,4,5\\right\\}\\)<\/p><p id=\"fs-id1167829593929\"><span class=\"token\">\u24d1<\/span> The range is the set of all <em data-effect=\"italics\">y<\/em>-values of the relation. \\(\\phantom{\\rule{4em}{0ex}}\\left\\{1,4,9,16,25\\right\\}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836629801\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833272898\"><div data-type=\"problem\" id=\"fs-id1167826169765\"><p>For the relation \\(\\left\\{\\left(1,1\\right),\\left(2,8\\right),\\left(3,27\\right),\\left(4,64\\right),\\left(5,125\\right)\\right\\}:\\)<\/p><p id=\"fs-id1167836550500\"><span class=\"token\">\u24d0<\/span> Find the domain of the relation.<\/p><p id=\"fs-id1167833202428\"><span class=\"token\">\u24d1<\/span> Find the range of the relation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836618853\"><p id=\"fs-id1167836492201\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{1,2,3,4,5\\right\\}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{1,8,27,64,125\\right\\}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833021555\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829614386\"><p id=\"fs-id1167836360520\">For the relation \\(\\left\\{\\left(1,3\\right),\\left(2,6\\right),\\left(3,9\\right),\\left(4,12\\right),\\left(5,15\\right)\\right\\}:\\)<\/p><p id=\"fs-id1167833290815\"><span class=\"token\">\u24d0<\/span> Find the domain of the relation.<\/p><p id=\"fs-id1167825702514\"><span class=\"token\">\u24d1<\/span> Find the range of the relation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829812976\"><p id=\"fs-id1167833346148\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{1,2,3,4,5\\right\\}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{3,6,9,12,15\\right\\}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836509402\"><div data-type=\"title\">Mapping<\/div><p id=\"fs-id1167824927876\">A <span data-type=\"term\">mapping<\/span> is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range.<\/p><\/div><div data-type=\"example\" id=\"fs-id1167829683746\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836525923\"><p id=\"fs-id1167833412571\">Use the <strong data-effect=\"bold\">mapping<\/strong> of the relation shown to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation, <span class=\"token\">\u24d1<\/span> find the domain of the relation, and <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><span data-type=\"media\" id=\"fs-id1167833364760\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cAlison\u201d, \u201cPenelope\u201d, \u201cJune\u201d, \u201cGregory\u201d, \u201cGeoffrey\u201d, \u201cLauren\u201d, \u201cStephen\u201d, \u201cAlice\u201d, \u201cLiz\u201d, \u201cDanny\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 12\u201d, \u201cFebruary 3\u201d, \u201cApril 25\u201d, \u201cMay 10\u201d, \u201cMay 23\u201d, \u201cJuly 24\u201d, \u201cAugust 2\u201d, and \u201cSeptember 15\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a date in the Birthday table. While most dates have only one arrow pointing to them, there are two arrows pointing to July 24: one from Stephen and one from Liz.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cAlison\u201d, \u201cPenelope\u201d, \u201cJune\u201d, \u201cGregory\u201d, \u201cGeoffrey\u201d, \u201cLauren\u201d, \u201cStephen\u201d, \u201cAlice\u201d, \u201cLiz\u201d, \u201cDanny\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 12\u201d, \u201cFebruary 3\u201d, \u201cApril 25\u201d, \u201cMay 10\u201d, \u201cMay 23\u201d, \u201cJuly 24\u201d, \u201cAugust 2\u201d, and \u201cSeptember 15\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a date in the Birthday table. While most dates have only one arrow pointing to them, there are two arrows pointing to July 24: one from Stephen and one from Liz.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167836417183\"><p id=\"fs-id1167836327324\"><span class=\"token\">\u24d0<\/span> The arrow shows the matching of the person to their birthday. We create ordered pairs with the person\u2019s name as the <em data-effect=\"italics\">x<\/em>-value and their birthday as the <em data-effect=\"italics\">y<\/em>-value.<\/p><p id=\"fs-id1167833008731\">{(Alison, April 25), (Penelope, May 23), (June, August 2), (Gregory, September 15), (Geoffrey, January 12), (Lauren, May 10), (Stephen, July 24), (Alice, February 3), (Liz, August 2), (Danny, July 24)}<\/p><p id=\"fs-id1167836788581\"><span class=\"token\">\u24d1<\/span> The domain is the set of all <em data-effect=\"italics\">x<\/em>-values of the relation.<\/p><p id=\"fs-id1167836648651\">{Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, Danny}<\/p><p id=\"fs-id1167836689824\"><span class=\"token\">\u24d2<\/span> The range is the set of all <em data-effect=\"italics\">y<\/em>-values of the relation.<\/p><p id=\"fs-id1167836326011\">{January 12, February 3, April 25, May 10, May 23, July 24, August 2, September 15}<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836340432\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829714243\"><p id=\"fs-id1167829614511\">Use the mapping of the relation shown to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><span data-type=\"media\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cKhanh Nguyen\u201d, \u201cAbigail Brown\u201d, \u201cSumantha Mishal\u201d, and \u201cJose Hern and ez\u201d. The table on the right has the header \u201cStudent ID #\u201d and lists the codes \u201ca b 56781\u201d, \u201cj h 47983\u201d, \u201ck n 68413\u201d, and \u201cs m 32479\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a code in the student ID table. The first arrow goes from Khanh Nguyen to k n 68413. The second arrow goes from Abigail Brown to a b 56781. The third arrow goes from Sumantha Mishal to s m 32479. The fourth arrow goes from Jose Hern and ez to j h 47983.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cKhanh Nguyen\u201d, \u201cAbigail Brown\u201d, \u201cSumantha Mishal\u201d, and \u201cJose Hern and ez\u201d. The table on the right has the header \u201cStudent ID #\u201d and lists the codes \u201ca b 56781\u201d, \u201cj h 47983\u201d, \u201ck n 68413\u201d, and \u201cs m 32479\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a code in the student ID table. The first arrow goes from Khanh Nguyen to k n 68413. The second arrow goes from Abigail Brown to a b 56781. The third arrow goes from Sumantha Mishal to s m 32479. The fourth arrow goes from Jose Hern and ez to j h 47983.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167836554322\"><p id=\"fs-id1167836539147\"><span class=\"token\">\u24d0<\/span> (Khanh Nguyen, kn68413), (Abigail Brown, ab56781), (Sumantha Mishal, sm32479), (Jose Hern and ez, jh47983) <span class=\"token\">\u24d1<\/span> {Khanh Nguyen, Abigail Brown, Sumantha Mishal, Jose Hern and ez} <span class=\"token\">\u24d2<\/span> {kn68413, ab56781, sm32479, jh47983}<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824754892\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836663646\"><div data-type=\"problem\" id=\"fs-id1167829716810\"><p id=\"fs-id1167836731398\">Use the mapping of the relation shown to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><span data-type=\"media\" id=\"fs-id1167833385478\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cMaria\u201d, \u201cArm and o\u201d, \u201cCynthia\u201d, \u201cKelly\u201d, and \u201cRachel\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 18\u201d, \u201cMarch 15\u201d, \u201cNovember 6\u201d, and \u201cDecember 8\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a date in the Birthday table. The first arrow goes from Maria to November 6. The second arrow goes from Arm and o to a January 18. The third arrow goes from Cynthia to December 8. The fourth arrow goes from Kelly to March 15. The fifth arrow goes from Rachel to November 6.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cMaria\u201d, \u201cArm and o\u201d, \u201cCynthia\u201d, \u201cKelly\u201d, and \u201cRachel\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 18\u201d, \u201cMarch 15\u201d, \u201cNovember 6\u201d, and \u201cDecember 8\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a date in the Birthday table. The first arrow goes from Maria to November 6. The second arrow goes from Arm and o to a January 18. The third arrow goes from Cynthia to December 8. The fourth arrow goes from Kelly to March 15. The fifth arrow goes from Rachel to November 6.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167829750633\"><p><span class=\"token\">\u24d0<\/span> (Maria, November 6), (Arm and o, January 18), (Cynthia, December 8), (Kelly, March 15), (Rachel, November 6) <span class=\"token\">\u24d1<\/span> {Maria, Arm and o, Cynthia, Kelly, Rachel} <span class=\"token\">\u24d2<\/span> {November 6, January 18, December 8, March 15}<\/p><\/div><\/div><\/div><p id=\"fs-id1167836532042\">A graph is yet another way that a relation can be represented. The set of ordered pairs of all the points plotted is the relation. The set of all <em data-effect=\"italics\">x<\/em>-coordinates is the domain of the relation and the set of all <em data-effect=\"italics\">y<\/em>-coordinates is the range. Generally we write the numbers in ascending order for both the domain and range.<\/p><div data-type=\"example\" id=\"fs-id1167833057329\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829877976\"><div data-type=\"problem\" id=\"fs-id1167836450184\"><p id=\"fs-id1167829681249\">Use the graph of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><span data-type=\"media\" id=\"fs-id1167836673420\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 1), (0, 3), (1, 5), (2, negative 2), and (4, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 1), (0, 3), (1, 5), (2, negative 2), and (4, negative 2).\"><\/span><\/div><div data-type=\"solution\"><p id=\"fs-id1167836613766\"><span class=\"token\">\u24d0<\/span> The ordered pairs of the relation are: \\(\\phantom{\\rule{7.6em}{0ex}}\\left\\{\\left(1,5\\right),\\left(-3,-1\\right),\\left(4,-2\\right),\\left(0,3\\right),\\left(2,-2\\right),\\left(-3,4\\right)\\right\\}.\\)<\/p><p id=\"fs-id1167836429436\"><span class=\"token\">\u24d1<\/span> The domain is the set of all <em data-effect=\"italics\">x<\/em>-values of the relation: \\(\\phantom{\\rule{2.2em}{0ex}}\\left\\{-3,0,1,2,4\\right\\}.\\)<\/p><p id=\"fs-id1167833137637\">Notice that while \\(-3\\) repeats, it is only listed once.<\/p><p id=\"fs-id1167833364739\"><span class=\"token\">\u24d2<\/span> The range is the set of all <em data-effect=\"italics\">y<\/em>-values of the relation: \\(\\phantom{\\rule{2.9em}{0ex}}\\left\\{-2,-1,3,4,5\\right\\}.\\)<\/p><p id=\"fs-id1167833050654\">Notice that while \\(-2\\) repeats, it is only listed once.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829750323\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829850985\"><div data-type=\"problem\" id=\"fs-id1167829879283\"><p id=\"fs-id1167824773962\">Use the graph of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><span data-type=\"media\" id=\"fs-id1167836341066\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 3), (negative 2, 2), (negative 1, 0), (0, negative 1), (2, negative 2), and (4, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 3), (negative 2, 2), (negative 1, 0), (0, negative 1), (2, negative 2), and (4, negative 4).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167836419079\"><p id=\"fs-id1167833051815\"><span class=\"token\">\u24d0<\/span>\\(\\left(-3,3\\right),\\left(-2,2\\right),\\left(-1,0\\right),\\)<\/p><div data-type=\"newline\"><br><\/div>\\(\\left(0,-1\\right),\\left(2,-2\\right),\\left(4,-4\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{-3,-2,-1,0,2,4\\right\\}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left\\{3,2,0,-1,-2,-4\\right\\}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829738598\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833056928\"><div data-type=\"problem\" id=\"fs-id1167836485775\"><p id=\"fs-id1167833050002\">Use the graph of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><span data-type=\"media\" id=\"fs-id1167833049956\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 5), (negative 3, 0), (negative 3, negative 6), (negative 1, negative 2), (1, 2), and (4, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 5), (negative 3, 0), (negative 3, negative 6), (negative 1, negative 2), (1, 2), and (4, negative 4).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167833021407\"><p id=\"fs-id1167836503993\"><span class=\"token\">\u24d0<\/span>\\(\\left(-3,0\\right),\\left(-3,5\\right),\\left(-3,-6\\right),\\)<\/p><div data-type=\"newline\"><br><\/div>\\(\\left(-1,-2\\right),\\left(1,2\\right),\\left(4,-4\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{-3,-1,1,4\\right\\}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\left\\{-6,0,5,-2,2,-4\\right\\}\\)<\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836610583\"><h3 data-type=\"title\">Determine if a Relation is a Function<\/h3><p id=\"fs-id1167826171759\">A special type of relation, called a <span data-type=\"term\">function<\/span>, occurs extensively in mathematics. A function 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><div data-type=\"note\" id=\"fs-id1167836614987\"><div data-type=\"title\">Function<\/div><p id=\"fs-id1167829714644\">A <strong data-effect=\"bold\">function<\/strong> is a relation that assigns to each element in its domain exactly one element in the range.<\/p><\/div><p id=\"fs-id1167836532939\">The birthday example from <a href=\"#fs-id1167829683746\" class=\"autogenerated-content\">(Figure)<\/a> helps us understand this definition. Every person has a birthday but no one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24<sup>th<\/sup> as their birthday and that June and Liz share August 2<sup>nd<\/sup>. Since each person has exactly one birthday, the relation in <a href=\"#fs-id1167829683746\" class=\"autogenerated-content\">(Figure)<\/a> is a function.<\/p><p id=\"fs-id1167836601426\">The relation shown by the graph in <a href=\"#fs-id1167833057329\" class=\"autogenerated-content\">(Figure)<\/a> includes the ordered pairs \\(\\left(-3,-1\\right)\\) and \\(\\left(-3,4\\right).\\) Is that okay in a function? No, as this is like one person having two different birthdays.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167824733092\"><div data-type=\"problem\" id=\"fs-id1167826132575\"><p id=\"fs-id1167836456488\">Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the relation.<\/p><p id=\"fs-id1167836518420\"><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\\}\\)<\/p><p id=\"fs-id1167824735617\"><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\left(9,-3\\right),\\left(4,-2\\right),\\left(1,-1\\right),\\left(0,0\\right),\\left(1,1\\right),\\left(4,2\\right),\\left(9,3\\right)\\right\\}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833052137\"><p id=\"fs-id1167836511265\"><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\\}\\)<\/p><p id=\"fs-id1167836691895\">(i) 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><p>(ii) The domain is the set of all <em data-effect=\"italics\">x<\/em>-values in the relation.<\/p><div data-type=\"newline\"><br><\/div>The domain is: \\(\\left\\{-3,-2,-1,0,1,2,3\\right\\}.\\)<p id=\"fs-id1167836282463\">(iii) The range is the set of all <em data-effect=\"italics\">y<\/em>-values in the relation. Notice we do not list range values twice.<\/p><div data-type=\"newline\"><br><\/div>The range is: \\(\\left\\{27,8,1,0\\right\\}.\\)<p id=\"fs-id1167824735962\"><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\left(9,-3\\right),\\left(4,-2\\right),\\left(1,-1\\right),\\left(0,0\\right),\\left(1,1\\right),\\left(4,2\\right),\\left(9,3\\right)\\right\\}\\)<\/p><p id=\"fs-id1167833223866\">(i) The <em data-effect=\"italics\">x<\/em>-value 9 is matched with two <em data-effect=\"italics\">y<\/em>-values, both 3 and \\(-3.\\) So this relation is not a function.<\/p><p>(ii) The domain is the set of all <em data-effect=\"italics\">x<\/em>-values in the relation. Notice we do not list domain values twice.<\/p><div data-type=\"newline\"><br><\/div>The domain is: \\(\\left\\{0,1,2,4,9\\right\\}.\\)<p id=\"fs-id1167836507848\">(iii) The range is the set of all <em data-effect=\"italics\">y<\/em>-values in the relation.<\/p><div data-type=\"newline\"><br><\/div>The range is: \\(\\left\\{-3,-2,-1,0,1,2,3\\right\\}.\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829742590\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836544009\"><p id=\"fs-id1167833060073\">Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the function.<\/p><p><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><p id=\"fs-id1167833380236\"><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\left(8,-4\\right),\\left(4,-2\\right),\\left(2,-1\\right),\\left(0,0\\right),\\left(2,1\\right),\\left(4,2\\right),\\left(8,4\\right)\\right\\}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836352889\"><p id=\"fs-id1167826188989\"><span class=\"token\">\u24d0<\/span> Yes; \\(\\left\\{-3,-2,-1,0,1,2,3\\right\\};\\)<\/p><div data-type=\"newline\"><br><\/div>\\(\\left\\{-6,-4,-2,0,2,4,6\\right\\}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> No; \\(\\left\\{0,2,4,8\\right\\};\\)<div data-type=\"newline\"><br><\/div>\\(\\left\\{-4,-2,-1,0,1,2,4\\right\\}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836362682\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836533244\"><div data-type=\"problem\" id=\"fs-id1167836717454\"><p id=\"fs-id1167836550940\">Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the relation.<\/p><p id=\"fs-id1167836598862\"><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\\}\\)<\/p><p id=\"fs-id1167836310540\"><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\\}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836298199\"><p id=\"fs-id1167836798080\"><span class=\"token\">\u24d0<\/span> No; \\(\\left\\{0,1,8,27\\right\\};\\)<\/p><div data-type=\"newline\"><br><\/div>\\(\\left\\{-3,-2,-1,0,2,2,3\\right\\}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Yes; \\(\\left\\{7,-5,8,0,-6,-2,-1\\right\\};\\)<div data-type=\"newline\"><br><\/div>\\(\\left\\{-3,-4,0,4,2,3\\right\\}\\)<\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167836539582\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829586801\"><div data-type=\"problem\" id=\"fs-id1167836684155\"><p id=\"fs-id1167833059664\">Use the mapping to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><span data-type=\"media\" id=\"fs-id1167829942558\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cLydia\u201d, \u201cEugene\u201d, \u201cJanet\u201d, \u201cRick\u201d, and \u201cMarty\u201d. The table on the right has the header \u201cPhone number\u201d and lists the numbers \u201c321-549-3327 home\u201d, \u201c427-658-2314 cell\u201d, \u201c321-964-7324 cell\u201d, \u201c684-358-7961 home\u201d, \u201c684-369-7231 cell\u201d, and \u201c798-367-8541 cell\u201d. There are arrows that start at a name and points toward a number in the phone number table. The first arrow goes from Lydia to 321-549-3327 home. The second arrow goes from Lydia to a 321-964-7324 cell. The third arrow goes from Eugene to 427-658-2314 cell. The fourth arrow goes from Janet to 427-658-2314 cell. The fifth arrow goes from Rick to 798-367-8541 cell. The sixth arrow goes from Marty to 684-358-7961 home. The seventh arrow goes from Marty to 684-369-7231 cell.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cLydia\u201d, \u201cEugene\u201d, \u201cJanet\u201d, \u201cRick\u201d, and \u201cMarty\u201d. The table on the right has the header \u201cPhone number\u201d and lists the numbers \u201c321-549-3327 home\u201d, \u201c427-658-2314 cell\u201d, \u201c321-964-7324 cell\u201d, \u201c684-358-7961 home\u201d, \u201c684-369-7231 cell\u201d, and \u201c798-367-8541 cell\u201d. There are arrows that start at a name and points toward a number in the phone number table. The first arrow goes from Lydia to 321-549-3327 home. The second arrow goes from Lydia to a 321-964-7324 cell. The third arrow goes from Eugene to 427-658-2314 cell. The fourth arrow goes from Janet to 427-658-2314 cell. The fifth arrow goes from Rick to 798-367-8541 cell. The sixth arrow goes from Marty to 684-358-7961 home. The seventh arrow goes from Marty to 684-369-7231 cell.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167836698471\"><p><span class=\"token\">\u24d0<\/span> Both Lydia and Marty have two phone numbers. So each <em data-effect=\"italics\">x<\/em>-value is not matched with only one <em data-effect=\"italics\">y<\/em>-value. So this relation is not a function.<\/p><p id=\"fs-id1167833369757\"><span class=\"token\">\u24d1<\/span> The domain is the set of all <em data-effect=\"italics\">x<\/em>-values in the relation. The domain is: {Lydia, Eugene, Janet, Rick, Marty}<\/p><p id=\"fs-id1167830076911\"><span class=\"token\">\u24d2<\/span> The range is the set of all <em data-effect=\"italics\">y<\/em>-values in the relation. The range is:<\/p><p id=\"fs-id1167836576219\">\\(\\left\\{321-549-3327,\\)\\(427-658-2314,\\)\\(321-964-7324,\\)\\(684-358-7961,\\)\\(684-369-7231,\\)\\(798-367-8541\\right\\}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836378848\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833397170\"><div data-type=\"problem\" id=\"fs-id1167836361517\"><p>Use the mapping to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><span data-type=\"media\" id=\"fs-id1167836392154\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNetwork\u201d and lists the television stations \u201cNBC\u201d, \u201cHGTV\u201d, and \u201cHBO\u201d. The table on the right has the header \u201cProgram\u201d and lists the television shows \u201cEllen Degeneres Show\u201d, \u201cLaw and Order\u201d, \u201cTonight Show\u201d, \u201cProperty Brothers\u201d, \u201cHouse Hunters\u201d, \u201cLove it or List it\u201d, \u201cGame of Thrones\u201d, \u201cTrue Detective\u201d, and \u201cSesame Street\u201d. There are arrows that start at a network in the first table and point toward a program in the second table. The first arrow goes from NBC to Ellen Degeneres Show. The second arrow goes from NBC to Law and Order. The third arrow goes from NBC to Tonight Show. The fourth arrow goes from HGTV to Property Brothers. The fifth arrow goes from HGTV to House Hunters. The sixth arrow goes from HGTV to Love it or List it. The seventh arrow goes from HBO to Game of Thrones. The eighth arrow goes from HBO to True Detective. The ninth arrow goes from HBO to Sesame Street.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNetwork\u201d and lists the television stations \u201cNBC\u201d, \u201cHGTV\u201d, and \u201cHBO\u201d. The table on the right has the header \u201cProgram\u201d and lists the television shows \u201cEllen Degeneres Show\u201d, \u201cLaw and Order\u201d, \u201cTonight Show\u201d, \u201cProperty Brothers\u201d, \u201cHouse Hunters\u201d, \u201cLove it or List it\u201d, \u201cGame of Thrones\u201d, \u201cTrue Detective\u201d, and \u201cSesame Street\u201d. There are arrows that start at a network in the first table and point toward a program in the second table. The first arrow goes from NBC to Ellen Degeneres Show. The second arrow goes from NBC to Law and Order. The third arrow goes from NBC to Tonight Show. The fourth arrow goes from HGTV to Property Brothers. The fifth arrow goes from HGTV to House Hunters. The sixth arrow goes from HGTV to Love it or List it. The seventh arrow goes from HBO to Game of Thrones. The eighth arrow goes from HBO to True Detective. The ninth arrow goes from HBO to Sesame Street.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167836388771\"><p id=\"fs-id1167829830534\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> {NBC, HGTV, HBO} <span class=\"token\">\u24d2<\/span> {Ellen Degeneres Show, Law and Order, Tonight Show, Property Brothers, House Hunters, Love it or List it, Game of Thrones, True Detective, Sesame Street}<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836511080\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836606033\"><div data-type=\"problem\" id=\"fs-id1167829579633\"><p id=\"fs-id1167833386907\">Use the mapping to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><span data-type=\"media\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cNeal\u201d, \u201cKrystal\u201d, \u201cKelvin\u201d, \u201cGeorge\u201d, \u201cChrista\u201d, and \u201cMike\u201d. The table on the right has the header \u201cPhone number\u201d and lists the numbers \u201c123-567-4389 work\u201d, \u201c231-378-5941 cell\u201d, \u201c753-469-9731 cell\u201d, \u201c567-534-2970 work\u201d, \u201c684-369-7231 cell\u201d, \u201c798-367-8541 cell\u201d, and \u201c639-847-6971 cell\u201d. There are arrows that start at a name and points toward a number in the phone number table. The first arrow goes from Neal to 753-469-9731 cell. The second arrow goes from Krystal to a 684-369-7231 cell. The third arrow goes from Kelvin to 231-378-5941 cell. The fourth arrow goes from George to 123-567-4389 work. The fifth arrow goes from George to 639-847-6971 cell. The sixth arrow goes from Christa to 567-534-2970 work. The seventh arrow goes from Mike to 567-534-2970 work. The eighth arrow goes from Mike to 798-367-8541 cell.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cNeal\u201d, \u201cKrystal\u201d, \u201cKelvin\u201d, \u201cGeorge\u201d, \u201cChrista\u201d, and \u201cMike\u201d. The table on the right has the header \u201cPhone number\u201d and lists the numbers \u201c123-567-4389 work\u201d, \u201c231-378-5941 cell\u201d, \u201c753-469-9731 cell\u201d, \u201c567-534-2970 work\u201d, \u201c684-369-7231 cell\u201d, \u201c798-367-8541 cell\u201d, and \u201c639-847-6971 cell\u201d. There are arrows that start at a name and points toward a number in the phone number table. The first arrow goes from Neal to 753-469-9731 cell. The second arrow goes from Krystal to a 684-369-7231 cell. The third arrow goes from Kelvin to 231-378-5941 cell. The fourth arrow goes from George to 123-567-4389 work. The fifth arrow goes from George to 639-847-6971 cell. The sixth arrow goes from Christa to 567-534-2970 work. The seventh arrow goes from Mike to 567-534-2970 work. The eighth arrow goes from Mike to 798-367-8541 cell.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167836754813\"><p id=\"fs-id1167836407181\"><span class=\"token\">\u24d0<\/span> No <span class=\"token\">\u24d1<\/span> {Neal, Krystal, Kelvin, George, Christa, Mike} <span class=\"token\">\u24d2<\/span> {123-567-4839 work, 231-378-5941 cell, 743-469-9731 cell, 567-534-2970 work, 684-369-7231 cell, 798-367-8541 cell, 639-847-6971 cell}<\/p><\/div><\/div><\/div><p id=\"fs-id1167836568027\">In algebra, more often than not, functions will be represented by an equation. It is easiest to see if the equation is a function when it is solved for <em data-effect=\"italics\">y<\/em>. If each value of <em data-effect=\"italics\">x<\/em> results in only one value of <em data-effect=\"italics\">y<\/em>, then the equation defines a function.<\/p><div data-type=\"example\" id=\"fs-id1167836683566\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836561265\"><div data-type=\"problem\" id=\"fs-id1167836506912\"><p id=\"fs-id1167829681109\">Determine whether each equation is a function.<\/p><p id=\"fs-id1167833303503\"><span class=\"token\">\u24d0<\/span>\\(2x+y=7\\)<span class=\"token\">\u24d1<\/span>\\(y={x}^{2}+1\\)<span class=\"token\">\u24d2<\/span>\\(x+{y}^{2}=3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836398976\"><p id=\"fs-id1167829599793\"><span class=\"token\">\u24d0<\/span>\\(2x+y=7\\)<\/p><p id=\"fs-id1167832926075\">For each value of <em data-effect=\"italics\">x<\/em>, we multiply it by \\(-2\\) and then add 7 to get the <em data-effect=\"italics\">y<\/em>-value<\/p><table class=\"unnumbered unstyled\" summary=\"y = negative 2 x plus 7. For example, if x = 3. y = negative 2 times 3 plus 7. y = 1.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836495341\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">For example, if \\(x=3:\\phantom{\\rule{2em}{0ex}}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836619694\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836326343\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167833056514\">We have that when \\(x=3,\\) then \\(y=1.\\) It would work similarly for any value of <em data-effect=\"italics\">x<\/em>. Since each value of <em data-effect=\"italics\">x<\/em>, corresponds to only one value of <em data-effect=\"italics\">y<\/em> the equation defines a function.<\/p><p id=\"fs-id1167836447354\"><span class=\"token\">\u24d1<\/span>\\(y={x}^{2}+1\\)<\/p><p id=\"fs-id1167829712213\">For each value of <em data-effect=\"italics\">x<\/em>, we square it and then add 1 to get the <em data-effect=\"italics\">y<\/em>-value.<\/p><table id=\"fs-id1167836533787\" class=\"unnumbered unstyled\" summary=\"y = x squared plus 1. For example, if x = 2. y = 2 squared plus 1. y = 5.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547666\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">For example, if \\(x=2:\\phantom{\\rule{2em}{0ex}}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836616301\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><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_03_05_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167829936815\">We have that when \\(x=2,\\) then \\(y=5.\\) It would work similarly for any value of <em data-effect=\"italics\">x<\/em>. Since each value of <em data-effect=\"italics\">x<\/em>, corresponds to only one value of <em data-effect=\"italics\">y<\/em> the equation defines a function.<\/p><p id=\"fs-id1167836515845\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829596595\" class=\"unnumbered unstyled\" summary=\"x plus y squared = 3. Isolate the y term. y squared = negative x plus 3. Let\u2019s substitute x = 2. y squared = negative 2 plus 3. y squared = 1.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829789899\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Isolate the <em data-effect=\"italics\">y<\/em> term.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833356461\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Let\u2019s substitute \\(x=2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836618915\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836296194\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td>This give us two values for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\">\\(y=1\\phantom{\\rule{0.2em}{0ex}}y=-1\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836539656\">We have shown that when \\(x=2,\\) then \\(y=1\\) and \\(y=-1.\\) It would work similarly for any value of <em data-effect=\"italics\">x<\/em>. Since each value of <em data-effect=\"italics\">x<\/em> does not corresponds to only one value of <em data-effect=\"italics\">y<\/em> the equation does not define a function.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833379684\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836585298\"><div data-type=\"problem\" id=\"fs-id1167836356048\"><p id=\"fs-id1167829716494\">Determine whether each equation is a function.<\/p><p id=\"fs-id1167836713610\"><span class=\"token\">\u24d0<\/span>\\(4x+y=-3\\)<span class=\"token\">\u24d1<\/span>\\(x+{y}^{2}=1\\)<span class=\"token\">\u24d2<\/span>\\(y-{x}^{2}=2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829811714\"><p id=\"fs-id1167833255910\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no <span class=\"token\">\u24d2<\/span> yes<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833369174\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836614714\"><p id=\"fs-id1167836730830\">Determine whether each equation is a function.<\/p><p id=\"fs-id1167836621950\"><span class=\"token\">\u24d0<\/span>\\(x+{y}^{2}=4\\)<span class=\"token\">\u24d1<\/span>\\(y={x}^{2}-7\\)<span class=\"token\">\u24d2<\/span>\\(y=5x-4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836289689\"><p id=\"fs-id1167829859295\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> yes<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167824731607\"><h3 data-type=\"title\">Find the Value of a Function<\/h3><p id=\"fs-id1167836315013\">It is very convenient to name a function and most often we name it <em data-effect=\"italics\">f<\/em>, <em data-effect=\"italics\">g<\/em>, <em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">F<\/em>, <em data-effect=\"italics\">G<\/em>, or <em data-effect=\"italics\">H<\/em>. In any function, for each <em data-effect=\"italics\">x<\/em>-value from the domain we get a corresponding <em data-effect=\"italics\">y<\/em>-value in the range. For the function <em data-effect=\"italics\">f<\/em>, we write this range value <em data-effect=\"italics\">y<\/em> as \\(f\\left(x\\right).\\) This is called function notation and is read <em data-effect=\"italics\">f<\/em> of <em data-effect=\"italics\">x<\/em> or the value of <em data-effect=\"italics\">f<\/em> at <em data-effect=\"italics\">x<\/em>. In this case the parentheses does not indicate multiplication.<\/p><div data-type=\"note\" id=\"fs-id1167824735502\"><div data-type=\"title\">Function Notation<\/div><p id=\"fs-id1167836379405\">For the function \\(y=f\\left(x\\right)\\)<\/p><div data-type=\"equation\" id=\"fs-id1167829590438\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}f\\phantom{\\rule{0.2em}{0ex}}\\text{is the name of the function}\\hfill \\\\ x\\phantom{\\rule{0.2em}{0ex}}\\text{is the domain value}\\hfill \\\\ f\\left(x\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{is the range value}\\phantom{\\rule{0.2em}{0ex}}y\\phantom{\\rule{0.2em}{0ex}}\\text{corresponding to the value}\\phantom{\\rule{0.2em}{0ex}}x\\hfill \\end{array}\\)<\/div><p>We read \\(f\\left(x\\right)\\) as <em data-effect=\"italics\">f<\/em> of <em data-effect=\"italics\">x<\/em> or the value of <em data-effect=\"italics\">f<\/em> at <em data-effect=\"italics\">x<\/em>.<\/p><\/div><p id=\"fs-id1167833186483\">We call <em data-effect=\"italics\">x<\/em> the independent variable as it can be any value in the domain. We call <em data-effect=\"italics\">y<\/em> the dependent variable as its value depends on <em data-effect=\"italics\">x<\/em>.<\/p><div data-type=\"note\" id=\"fs-id1167829620795\"><div data-type=\"title\">Independent and Dependent Variables<\/div><p id=\"fs-id1167836548512\">For the function \\(y=f\\left(x\\right),\\)<\/p><div data-type=\"equation\" id=\"fs-id1167836732021\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}x\\phantom{\\rule{0.2em}{0ex}}\\text{is the independent variable as it can be any value in the domain}\\hfill \\\\ y\\phantom{\\rule{0.2em}{0ex}}\\text{the dependent variable as its value depends on}\\phantom{\\rule{0.2em}{0ex}}x\\hfill \\end{array}\\)<\/div><\/div><p id=\"fs-id1167836362234\">Much as when you first encountered the variable <em data-effect=\"italics\">x<\/em>, function notation may be rather unsettling. It seems strange because it is new. You will feel more comfortable with the notation as you use it.<\/p><p id=\"fs-id1167833051398\">Let\u2019s look at the equation \\(y=4x-5.\\) To find the value of <em data-effect=\"italics\">y<\/em> when \\(x=2,\\) we know to substitute \\(x=2\\) into the equation and then simplify.<\/p><table class=\"unnumbered unstyled\" summary=\"y = 4 x minus 5. Let x = 2. y = 4 times 2 minus 5. y = 3.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836353205\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Let \\(x=2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836433912\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832926879\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167829718470\">The value of the function at \\(x=2\\) is 3.<\/p><p id=\"fs-id1167836560949\">We do the same thing using function notation, the equation \\(y=4x-5\\) can be written as \\(f\\left(x\\right)=4x-5.\\) To find the value when \\(x=2,\\) we write:<\/p><table id=\"fs-id1167836326537\" class=\"unnumbered unstyled\" summary=\"f of x = 4 x minus 5. Let x = 2. f of 2 = 4 times 2 minus 5. f of 2 = 3.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836447887\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Let \\(x=2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836314769\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836399865\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836456485\">The value of the function at \\(x=2\\) is 3.<\/p><p id=\"fs-id1167836629682\">This process of finding the value of \\(f\\left(x\\right)\\) for a given value of <em data-effect=\"italics\">x<\/em> is called <em data-effect=\"italics\">evaluating the function.<\/em><\/p><div data-type=\"example\" id=\"fs-id1167836521479\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836596680\"><div data-type=\"problem\" id=\"fs-id1167833224521\"><p id=\"fs-id1167833369207\">For the function \\(f\\left(x\\right)=2{x}^{2}+3x-1,\\) evaluate the function.<\/p><p id=\"fs-id1167836416554\"><span class=\"token\">\u24d0<\/span>\\(f\\left(3\\right)\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(-2\\right)\\)<span class=\"token\">\u24d2<\/span>\\(f\\left(a\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829756296\"><p id=\"fs-id1167829590752\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836507536\" class=\"unnumbered unstyled\" summary=\"f of x = 2 x squared plus 3 x minus 1. To evaluate f of 3, substitute 3 for x. f of 3 = 2 times 3 squared plus 3 times 3 minus 1. Simplify. f of 3 = 2 times 9 plus 3 times 3 minus 1. f of 3 = 18 plus 9 minus 1. f of 3 = 26.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836293439\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">To evaluate \\(f\\left(3\\right),\\) substitute 3 for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832999716\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836520086\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832980523\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829852974\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836775152\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829740069\" class=\"unnumbered unstyled\" summary=\"f of x = 2 x squared plus 3 x minus 1. To evaluate f of negative 2, substitute negative 2 for x. f of negative 2 = 2 times the quantity negative 2 in parentheses squared plus 3 times negative 2 minus 1. Simplify. f of negative 2 = 2 times 4 plus negative 6 minus 1. f of negative 2 = 1.\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><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_03_05_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792372816\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829694507\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829715540\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836532685\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829695370\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836601841\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836600305\" class=\"unnumbered unstyled\" summary=\"f of x = 2 x squared plus 3 x minus 1. To evaluate f of a, substitute a for x. f of a = 2 times a squared plus 3 times a minus 1. Simplify. f of a = 2 a squared plus 3 a minus 1.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836685384\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_017a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">To evaluate \\(f\\left(a\\right),\\) substitute <em data-effect=\"italics\">a<\/em> for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836362257\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836320903\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836388368\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836648690\"><div data-type=\"problem\" id=\"fs-id1167836520515\"><p id=\"fs-id1167836363586\">For the function \\(f\\left(x\\right)=3{x}^{2}-2x+1,\\) evaluate the function.<\/p><p id=\"fs-id1167836625930\"><span class=\"token\">\u24d0<\/span>\\(f\\left(3\\right)\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(-1\\right)\\)<span class=\"token\">\u24d2<\/span>\\(f\\left(t\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833135306\"><p id=\"fs-id1167825884792\"><span class=\"token\">\u24d0<\/span>\\(f\\left(3\\right)=22\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(-1\\right)=6\\)<span class=\"token\">\u24d2<\/span>\\(f\\left(t\\right)=3{t}^{2}-2t-1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824578714\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836729386\"><div data-type=\"problem\" id=\"fs-id1167833269886\"><p id=\"fs-id1167833269888\">For the function \\(f\\left(x\\right)=2{x}^{2}+4x-3,\\) evaluate the function.<\/p><p id=\"fs-id1167829579645\"><span class=\"token\">\u24d0<\/span>\\(f\\left(2\\right)\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(-3\\right)\\)<span class=\"token\">\u24d2<\/span>\\(f\\left(h\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829783753\"><p id=\"fs-id1167836732807\"><span class=\"token\">\u24d0<\/span>\\(\\left(2\\right)=13\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(-3\\right)=3\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(f\\left(h\\right)=2{h}^{2}+4h-3\\)<\/div><\/div><\/div><p id=\"fs-id1167836518536\">In the last example, we found \\(f\\left(x\\right)\\) for a constant value of <em data-effect=\"italics\">x<\/em>. In the next example, we are asked to find \\(g\\left(x\\right)\\) with values of <em data-effect=\"italics\">x<\/em> that are variables. We still follow the same procedure and substitute the variables in for the <em data-effect=\"italics\">x<\/em>.<\/p><div data-type=\"example\" id=\"fs-id1167829859398\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833137652\"><div data-type=\"problem\" id=\"fs-id1167836619820\"><p id=\"fs-id1167836619822\">For the function \\(g\\left(x\\right)=3x-5,\\) evaluate the function.<\/p><p id=\"fs-id1167836320931\"><span class=\"token\">\u24d0<\/span>\\(g\\left({h}^{2}\\right)\\)<span class=\"token\">\u24d1<\/span>\\(g\\left(x+2\\right)\\)<span class=\"token\">\u24d2<\/span>\\(g\\left(x\\right)+g\\left(2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833397154\"><p id=\"fs-id1167836754871\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1171790386499\" class=\"unnumbered unstyled\" summary=\"g of x = 3 x minus 5. To evaluate g of h squared, substitute h squared for x. g of h squared = 3 times h squared minus 5. Simplify. g of h squared = 3 h squared minus 5.\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792580809\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_018a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">To evaluate \\(g\\left({h}^{2}\\right),\\) substitute \\({h}^{2}\\) for <em data-effect=\"italics\">x<\/em>.<\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171790163372\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_018b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792588637\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_018c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836556251\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1171792580965\" class=\"unnumbered unstyled\" summary=\"g of x = 3 x minus 5. To evaluate g of the quantity x plus 2, substitute x plus 2 for x. g of x plus 2 = 3 times the quantity x plus 2 in parentheses minus 5. Simplify. g of x plus 2 = 3 x plus 6 minus 5. g of x plus 2 = 3 x plus 1.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792545190\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_019a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">To evaluate \\(g\\left(x+2\\right),\\) substitute \\(x+2\\) for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792543055\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_019b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792802121\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_019c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171790304229\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_019d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836448921\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836606104\" class=\"unnumbered unstyled\" summary=\"g of x = 3 x minus 5. To evaluate g of x plus g of 2, first find g of 2. g of 2 = 3 times 2 minus 5. g of 2 = 1. Now find g of x plus g of 2. g of x plus g of 2 = 3 x minus 5 plus 1. The 3 x minus 5 is from g of x and the 1 is from the g of 2. Simplify. g of x plus g of 2 = 3 x minus 4.\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829839522\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">To evaluate \\(g\\left(x\\right)+g\\left(2\\right),\\) first find \\(g\\left(2\\right).\\)<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836635207\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829685715\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836693279\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836322927\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td><\/td><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_03_05_020f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836539801\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167832981642\">Notice the difference between part <span class=\"token\">\u24d1<\/span> and <span class=\"token\">\u24d2<\/span>. We get \\(g\\left(x+2\\right)=3x+1\\) and \\(g\\left(x\\right)+g\\left(2\\right)=3x-4.\\) So we see that \\(g\\left(x+2\\right)\\ne g\\left(x\\right)+g\\left(2\\right).\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836730594\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832940643\"><div data-type=\"problem\" id=\"fs-id1167829788685\"><p id=\"fs-id1167829788687\">For the function \\(g\\left(x\\right)=4x-7,\\) evaluate the function.<\/p><p id=\"fs-id1167829906594\"><span class=\"token\">\u24d0<\/span>\\(g\\left({m}^{2}\\right)\\)<span class=\"token\">\u24d1<\/span>\\(g\\left(x-3\\right)\\)<span class=\"token\">\u24d2<\/span>\\(g\\left(x\\right)-g\\left(3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836624292\"><p id=\"fs-id1167836554185\"><span class=\"token\">\u24d0<\/span>\\(4{m}^{2}-7\\)<span class=\"token\">\u24d1<\/span>\\(4x-19\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(x-12\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167825766170\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829850927\"><p id=\"fs-id1167836293432\">For the function \\(h\\left(x\\right)=2x+1,\\) evaluate the function.<\/p><p id=\"fs-id1167823012018\"><span class=\"token\">\u24d0<\/span>\\(h\\left({k}^{2}\\right)\\)<span class=\"token\">\u24d1<\/span>\\(h\\left(x+1\\right)\\)<span class=\"token\">\u24d2<\/span>\\(h\\left(x\\right)+h\\left(1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829743779\"><p id=\"fs-id1167829717717\"><span class=\"token\">\u24d0<\/span>\\(2{k}^{2}+1\\)<span class=\"token\">\u24d1<\/span>\\(2x+3\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(2x+4\\)<\/div><\/div><\/div><p id=\"fs-id1167836283159\">Many everyday situations can be modeled using functions.<\/p><div data-type=\"example\" id=\"fs-id1167833158753\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833158755\"><div data-type=\"problem\" id=\"fs-id1167836533836\"><p id=\"fs-id1167836533838\">The number of unread emails in Sylvia\u2019s account is 75. This number grows by 10 unread emails a day. The function \\(N\\left(t\\right)=75+10t\\) represents the relation between the number of emails, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in days.<\/p><p id=\"fs-id1167836442406\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p><p id=\"fs-id1167833061546\"><span class=\"token\">\u24d1<\/span> Find \\(N\\left(5\\right).\\) Explain what this result means.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829715383\"><p id=\"fs-id1167829715385\"><span class=\"token\">\u24d0<\/span> The number of unread emails is a function of the number of days. The number of unread emails, <em data-effect=\"italics\">N<\/em>, depends on the number of days, <em data-effect=\"italics\">t<\/em>. Therefore, the variable <em data-effect=\"italics\">N<\/em>, is the dependent variable and the variable \\(t\\) is the independent variable.<\/p><p id=\"fs-id1167833347223\"><span class=\"token\">\u24d1<\/span> Find \\(N\\left(5\\right).\\) Explain what this result means.<\/p><table id=\"fs-id1167829709312\" class=\"unnumbered unstyled\" summary=\"N of t = 75 plus 10 t. Substitute in t = 5. N of 5 = 75 plus 10 times 5. Simplify. N of 5 = 75 plus 50. N of 5 = 125.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829850506\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Substitute in \\(t=5.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832971244\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_021b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833270224\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_021c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833309949\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_021d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167833413059\">Since 5 is the number of days, \\(N\\left(5\\right),\\) is the number of unread emails after 5 days. After 5 days, there are 125 unread emails in the account.<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836507876\"><p id=\"fs-id1167836507878\">The number of unread emails in Bryan\u2019s account is 100. This number grows by 15 unread emails a day. The function \\(N\\left(t\\right)=100+15t\\) represents the relation between the number of emails, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in days.<\/p><p id=\"fs-id1167836360895\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p><p id=\"fs-id1167829619314\"><span class=\"token\">\u24d1<\/span> Find \\(N\\left(7\\right).\\) Explain what this result means.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836622270\"><p id=\"fs-id1167836622272\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">t<\/em> IND; <em data-effect=\"italics\">N<\/em> DEP <span class=\"token\">\u24d1<\/span> 205; the number of unread emails in Bryan\u2019s account on the seventh day.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836481611\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829786754\"><div data-type=\"problem\" id=\"fs-id1167824735581\"><p id=\"fs-id1167824735583\">The number of unread emails in Anthony\u2019s account is 110. This number grows by 25 unread emails a day. The function \\(N\\left(t\\right)=110+25t\\) represents the relation between the number of emails, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in days.<\/p><p id=\"fs-id1167829743199\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p><p id=\"fs-id1167836575515\"><span class=\"token\">\u24d1<\/span> Find \\(N\\left(14\\right).\\) Explain what this result means.<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836656679\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">t<\/em> IND; <em data-effect=\"italics\">N<\/em> DEP <span class=\"token\">\u24d1<\/span> 460; the number of unread emails in Anthony\u2019s account on the fourteenth day<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833128952\" class=\"media-2\"><p id=\"fs-id1167836508744\">Access this online resource for additional instruction and practice with relations and functions.<\/p><ul id=\"fs-id1167836508748\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37introfunction\">Introduction to Functions<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829711772\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167836294674\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Function Notation:<\/strong> For the function \\(y=f\\left(x\\right)\\) <ul id=\"fs-id1167829810537\" data-bullet-style=\"open-circle\"><li><em data-effect=\"italics\">f<\/em> is the name of the function<\/li><li><em data-effect=\"italics\">x<\/em> is the domain value<\/li><li>\\(f\\left(x\\right)\\) is the range value <em data-effect=\"italics\">y<\/em> corresponding to the value <em data-effect=\"italics\">x<\/em><div data-type=\"newline\"><br><\/div> We read \\(f\\left(x\\right)\\) as <em data-effect=\"italics\">f<\/em> of <em data-effect=\"italics\">x<\/em> or the value of <em data-effect=\"italics\">f<\/em> at <em data-effect=\"italics\">x<\/em>.<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Independent and Dependent Variables:<\/strong> For the function \\(y=f\\left(x\\right),\\) <ul id=\"fs-id1167836715045\" data-bullet-style=\"open-circle\"><li><em data-effect=\"italics\">x<\/em> is the independent variable as it can be any value in the domain<\/li><li><em data-effect=\"italics\">y<\/em> is the dependent variable as its value depends on <em data-effect=\"italics\">x<\/em><\/li><\/ul><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167826170977\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167826189010\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167833041789\"><strong data-effect=\"bold\">Find the Domain and Range of a Relation<\/strong><\/p><p id=\"fs-id1167836701331\">In the following exercises, for each relation <span class=\"token\">\u24d0<\/span> find the domain of the relation <span class=\"token\">\u24d1<\/span> find the range of the relation.<\/p><div data-type=\"exercise\" id=\"fs-id1167836694560\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833129254\"><p id=\"fs-id1167833129256\">\\(\\left\\{\\left(1,4\\right),\\left(2,8\\right),\\left(3,12\\right),\\left(4,16\\right),\\left(5,20\\right)\\right\\}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836615948\"><p id=\"fs-id1167836615950\"><span class=\"token\">\u24d0<\/span> {1, 2, 3, 4, 5} <span class=\"token\">\u24d1<\/span> {4, 8, 12, 16, 20}<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829738657\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836729077\"><p id=\"fs-id1167836729080\">\\(\\left\\{\\left(1,-2\\right),\\left(2,-4\\right),\\left(3,-6\\right),\\left(4,-8\\right),\\left(5,-10\\right)\\right\\}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836289174\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836289176\"><p id=\"fs-id1167833021591\">\\(\\left\\{\\left(1,7\\right),\\left(5,3\\right),\\left(7,9\\right),\\left(-2,-3\\right),\\left(-2,8\\right)\\right\\}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829713369\"><p id=\"fs-id1167829713371\"><span class=\"token\">\u24d0<\/span> {1, 5, 7, \u22122} <span class=\"token\">\u24d1<\/span> {7, 3, 9, \u22123, 8}<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836513003\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836686245\"><p id=\"fs-id1167836686247\">\\(\\left\\{\\left(11,3\\right),\\left(-2,-7\\right),\\left(4,-8\\right),\\left(4,17\\right),\\left(-6,9\\right)\\right\\}\\)<\/p><\/div><\/div><p id=\"fs-id1167836309438\">In the following exercises, use the mapping of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation, <span class=\"token\">\u24d1<\/span> find the domain of the relation, and <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><div data-type=\"exercise\" id=\"fs-id1167824736897\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167829693414\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167829693415\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cRebecca\u201d, \u201cJennifer\u201d, \u201cJohn\u201d, \u201cHector\u201d, \u201cLuis\u201d, \u201cEbony\u201d, \u201cRaphael\u201d, \u201cMeredith\u201d, \u201cKaren\u201d, and \u201cJoseph\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 18\u201d, \u201cFebruary 15\u201d, \u201cApril 1\u201d, \u201cApril 7\u201d, \u201cJune 23\u201d, \u201cJuly 30\u201d, \u201cAugust 19\u201d, and \u201cNovember 6\u201d. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Rebecca to January 18. The second arrow goes from Jennifer to April 1. The third arrow goes from John to January 18. The fourth arrow goes from Hector to June 23. The fifth arrow goes from Luis to February 15. The sixth arrow goes from Ebony to April 7. The seventh arrow goes from Raphael to November 6. The eighth arrow goes from Meredith to August 19. The ninth arrow goes from Karen to August 19. The tenth arrow goes from Joseph to July 30.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cRebecca\u201d, \u201cJennifer\u201d, \u201cJohn\u201d, \u201cHector\u201d, \u201cLuis\u201d, \u201cEbony\u201d, \u201cRaphael\u201d, \u201cMeredith\u201d, \u201cKaren\u201d, and \u201cJoseph\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 18\u201d, \u201cFebruary 15\u201d, \u201cApril 1\u201d, \u201cApril 7\u201d, \u201cJune 23\u201d, \u201cJuly 30\u201d, \u201cAugust 19\u201d, and \u201cNovember 6\u201d. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Rebecca to January 18. The second arrow goes from Jennifer to April 1. The third arrow goes from John to January 18. The fourth arrow goes from Hector to June 23. The fifth arrow goes from Luis to February 15. The sixth arrow goes from Ebony to April 7. The seventh arrow goes from Raphael to November 6. The eighth arrow goes from Meredith to August 19. The ninth arrow goes from Karen to August 19. The tenth arrow goes from Joseph to July 30.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167836558109\"><p id=\"fs-id1167833369090\"><span class=\"token\">\u24d0<\/span> (Rebecca, January 18), (Jennifer, April 1), (John, January 18), (Hector, June 23), (Luis, February 15), (Ebony, April 7), (Raphael, November 6), (Meredith, August 19), (Karen, August 19), (Joseph, July 30)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> {Rebecca, Jennifer, John, Hector, Luis, Ebony, Raphael, Meredith, Karen, Joseph}<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> {January 18, April 1, June 23, February 15, April 7, November 6, August 19, July 30}<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836600990\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829614614\"><p id=\"fs-id1167829614616\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167829614618\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cAmy\u201d, \u201cCarol\u201d, \u201cDevon\u201d, \u201cHarrison\u201d, \u201cJackson\u201d, \u201cLabron\u201d, \u201cMason\u201d, \u201cNatalie\u201d, \u201cPaul\u201d, and \u201cSylvester\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 5\u201d, \u201cJanuary 7\u201d, \u201cFebruary 14\u201d, \u201cMarch 1\u201d, \u201cApril 7\u201d, \u201cMay 30\u201d, \u201cJuly 20\u201d, \u201cAugust 1\u201d, \u201cNovember 13\u201d, and \u201cNovember 26\u201d. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Amy to February 14. The second arrow goes from Carol to May 30. The third arrow goes from Devon to January 5. The fourth arrow goes from Harrison to January 7. The fifth arrow goes from Jackson to November 26. The sixth arrow goes from Labron to April 7. The seventh arrow goes from Mason to July 20. The eighth arrow goes from Natalie to March 1. The ninth arrow goes from Paul to August 1. The tenth arrow goes from Sylvester to November 13.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cAmy\u201d, \u201cCarol\u201d, \u201cDevon\u201d, \u201cHarrison\u201d, \u201cJackson\u201d, \u201cLabron\u201d, \u201cMason\u201d, \u201cNatalie\u201d, \u201cPaul\u201d, and \u201cSylvester\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 5\u201d, \u201cJanuary 7\u201d, \u201cFebruary 14\u201d, \u201cMarch 1\u201d, \u201cApril 7\u201d, \u201cMay 30\u201d, \u201cJuly 20\u201d, \u201cAugust 1\u201d, \u201cNovember 13\u201d, and \u201cNovember 26\u201d. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Amy to February 14. The second arrow goes from Carol to May 30. The third arrow goes from Devon to January 5. The fourth arrow goes from Harrison to January 7. The fifth arrow goes from Jackson to November 26. The sixth arrow goes from Labron to April 7. The seventh arrow goes from Mason to July 20. The eighth arrow goes from Natalie to March 1. The ninth arrow goes from Paul to August 1. The tenth arrow goes from Sylvester to November 13.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833128978\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833128980\"><p id=\"fs-id1167836511320\">For a woman of height \\(5\\prime 4\u2033\\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \\(18.5\u201324.9\\) is considered healthy.<\/p><span data-type=\"media\" id=\"fs-id1167829850494\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cWeight (lbs)\u201d and lists the numbers plus 100, 110, 120, 130, 140, 150, and 160. The table on the right has the header \u201cBMI\u201d and lists the numbers 18. 9, 22. 3, 17. 2, 24. 0, 25. 7, 20. 6, and 27. 5. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from plus 100 to 17. 2. The second arrow goes from 110 to 18. 9. The third arrow goes from 120 to 20. 6. The fourth arrow goes from 130 to 22. 3. The fifth arrow goes from 140 to 24. 0. The sixth arrow goes from 150 to 25. 7. The seventh arrow goes from 160 to 27. 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cWeight (lbs)\u201d and lists the numbers plus 100, 110, 120, 130, 140, 150, and 160. The table on the right has the header \u201cBMI\u201d and lists the numbers 18. 9, 22. 3, 17. 2, 24. 0, 25. 7, 20. 6, and 27. 5. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from plus 100 to 17. 2. The second arrow goes from 110 to 18. 9. The third arrow goes from 120 to 20. 6. The fourth arrow goes from 130 to 22. 3. The fifth arrow goes from 140 to 24. 0. The sixth arrow goes from 150 to 25. 7. The seventh arrow goes from 160 to 27. 5.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167836410493\"><p id=\"fs-id1167836410496\"><span class=\"token\">\u24d0<\/span> (+100, 17. 2), (110, 18.9), (120, 20.6), (130, 22.3), (140, 24.0), (150, 25.7), (160, 27.5) <span class=\"token\">\u24d1<\/span> {+100, 110, 120, 130, 140, 150, 160,} <span class=\"token\">\u24d2<\/span> {17.2, 18.9, 20.6, 22.3, 24.0, 25.7, 27.5}<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829590523\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829590525\"><p id=\"fs-id1167829590527\">For a man of height \\(5\\prime 11\\prime \\prime \\) the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of \\(18.5\u201324.9\\) is considered healthy.<\/p><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167836328549\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cWeight (lbs)\u201d and lists the numbers 130, 140, 150, 160, 170, 180, 190, and 200. The table on the right has the header \u201cBMI\u201d and lists the numbers 22. 3, 19. 5, 20. 9, 27. 9, 25. 1, 26. 5, 23. 7, and 18. 1. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from 130 to 18. 1. The second arrow goes from 140 to 19. 5. The third arrow goes from 150 to 20. 9. The fourth arrow goes from 160 to 22. 3. The fifth arrow goes from 170 to 23. 7. The sixth arrow goes from 180 to 25. 1. The seventh arrow goes from 190 to 26. 5. The eighth arrow goes from 200 to 27. 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cWeight (lbs)\u201d and lists the numbers 130, 140, 150, 160, 170, 180, 190, and 200. The table on the right has the header \u201cBMI\u201d and lists the numbers 22. 3, 19. 5, 20. 9, 27. 9, 25. 1, 26. 5, 23. 7, and 18. 1. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from 130 to 18. 1. The second arrow goes from 140 to 19. 5. The third arrow goes from 150 to 20. 9. The fourth arrow goes from 160 to 22. 3. The fifth arrow goes from 170 to 23. 7. The sixth arrow goes from 180 to 25. 1. The seventh arrow goes from 190 to 26. 5. The eighth arrow goes from 200 to 27. 9.\"><\/span><\/div><\/div><p id=\"fs-id1167825791209\">In the following exercises, use the graph of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><div data-type=\"exercise\" id=\"fs-id1167836621459\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836623116\"><p id=\"fs-id1167836623118\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836623119\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 1), (0, negative 3), (2, 3), (4, negative 1), and (4, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 1), (0, negative 3), (2, 3), (4, negative 1), and (4, negative 3).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167836448402\"><p id=\"fs-id1167836448404\"><span class=\"token\">\u24d0<\/span> (2, 3), (4, \u22123), (\u22122, \u22121), (\u22123, 4), (4, \u22121), (0, \u22123) <span class=\"token\">\u24d1<\/span> {\u22123, \u22122, 0, 2, 4}<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> {\u22123, \u22121, 3, 4}<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833274699\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833274701\"><p id=\"fs-id1167836599639\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836599640\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 4), (negative 2, 0), (negative 1, 3), (1, 5), and (4, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 4), (negative 2, 0), (negative 1, 3), (1, 5), and (4, negative 2).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836707143\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836524480\"><p id=\"fs-id1167836524482\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836524483\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 1, 4), (negative 1, negative 4), (0, 3), (0, negative 3), (1, 4), and (1, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_207_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 1, 4), (negative 1, negative 4), (0, 3), (0, negative 3), (1, 4), and (1, negative 4).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167829861860\"><p id=\"fs-id1167829861862\"><span class=\"token\">\u24d0<\/span> (1, 4), (1, \u22124), (\u22121, 4), (\u22121, \u22124), (0, 3), (0, \u22123) <span class=\"token\">\u24d1<\/span> {\u22121, 0, 1} <span class=\"token\">\u24d2<\/span> {\u22124, \u22123, 3,4}<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836509162\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836509164\"><p id=\"fs-id1167836546295\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836546296\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The points (negative 2, negative 6), (negative 2, negative 3), (0, 0), (0. 5, 1. 5), (1, 3), and (3, 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_208_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The points (negative 2, negative 6), (negative 2, negative 3), (0, 0), (0. 5, 1. 5), (1, 3), and (3, 6).\"><\/span><\/div><\/div><p id=\"fs-id1167836513559\"><strong data-effect=\"bold\">Determine if a Relation is a Function<\/strong><\/p><p id=\"fs-id1167833060894\">In the following exercises, use the set of ordered pairs to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function, <span class=\"token\">\u24d1<\/span> find the domain of the relation, and <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p><div data-type=\"exercise\" id=\"fs-id1167829666517\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833059308\"><p id=\"fs-id1167833059310\">\\(\\left\\{\\left(-3,9\\right),\\left(-2,4\\right),\\left(-1,1\\right),\\)<\/p><div data-type=\"newline\"><br><\/div>\\(\\left(0,0\\right),\\left(1,1\\right),\\left(2,4\\right),\\left(3,9\\right)\\right\\}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167833137411\"><p id=\"fs-id1167833137413\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> {\u22123, \u22122, \u22121, 0, 1, 2, 3} <span class=\"token\">\u24d2<\/span> {9, 4, 1, 0}<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836688996\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836688998\"><p id=\"fs-id1167836689000\">\\(\\left\\{\\left(9,-3\\right),\\left(4,-2\\right),\\left(1,-1\\right),\\)<\/p><div data-type=\"newline\"><br><\/div>\\(\\left(0,0\\right),\\left(1,1\\right),\\left(4,2\\right),\\left(9,3\\right)\\right\\}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829745706\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829745708\"><p id=\"fs-id1167829745710\">\\(\\left\\{\\left(-3,27\\right),\\left(-2,8\\right),\\left(-1,1\\right),\\)<\/p><div data-type=\"newline\"><br><\/div>\\(\\left(0,0\\right),\\left(1,1\\right),\\left(2,8\\right),\\left(3,27\\right)\\right\\}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836529454\"><p id=\"fs-id1167829853364\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> {\u22123, \u22122, \u22121, 0, 1, 2, 3} <span class=\"token\">\u24d2<\/span> 0, 1, 8, 27}<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833138708\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833138710\"><p id=\"fs-id1167826077064\">\\(\\left\\{\\left(-3,-27\\right),\\left(-2,-8\\right),\\left(-1,-1\\right),\\)<\/p><div data-type=\"newline\"><br><\/div>\\(\\left(0,0\\right),\\left(1,1\\right),\\left(2,8\\right),\\left(3,27\\right)\\right\\}\\)<\/div><\/div><p id=\"fs-id1167836552752\">In the following exercises, use the mapping to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function, <span class=\"token\">\u24d1<\/span> find the domain of the function, and <span class=\"token\">\u24d2<\/span> find the range of the function.<\/p><div data-type=\"exercise\" id=\"fs-id1167829742787\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829742789\"><p id=\"fs-id1167829742791\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167833056754\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNumber\u201d and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header \u201cAbsolute Value\u201d and lists the numbers 0, 1, 2, and 3. There are arrows starting at numbers in the number table and pointing towards numbers in the absolute value table. The first arrow goes from negative 3 to 3. The second arrow goes from negative 2 to 2. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 2. The seventh arrow goes from 3 to 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_209_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNumber\u201d and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header \u201cAbsolute Value\u201d and lists the numbers 0, 1, 2, and 3. There are arrows starting at numbers in the number table and pointing towards numbers in the absolute value table. The first arrow goes from negative 3 to 3. The second arrow goes from negative 2 to 2. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 2. The seventh arrow goes from 3 to 3.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167829749063\"><p id=\"fs-id1167829749065\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> {\u22123, \u22122, \u22121, 0, 1, 2, 3} <span class=\"token\">\u24d2<\/span> {0, 1, 2, 3}<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829741614\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829741616\"><p id=\"fs-id1167829908273\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167829908274\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNumber\u201d and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header \u201cSquare\u201d and lists the numbers 0, 1, 4, and 9. There are arrows starting at numbers in the number table and pointing towards numbers in the square table. The first arrow goes from negative 3 to 9. The second arrow goes from negative 2 to 4. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 4. The seventh arrow goes from 3 to 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_210_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNumber\u201d and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header \u201cSquare\u201d and lists the numbers 0, 1, 4, and 9. There are arrows starting at numbers in the number table and pointing towards numbers in the square table. The first arrow goes from negative 3 to 9. The second arrow goes from negative 2 to 4. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 4. The seventh arrow goes from 3 to 9.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836697708\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836697710\"><p id=\"fs-id1167832930330\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cJenny\u201d, \u201cR and y\u201d, \u201cDennis\u201d, \u201cEmily\u201d, and \u201cRaul\u201d. The table on the right has the header \u201cEmail\u201d and lists the email addresses RHern and ez@state. edu, JKim@gmail.com, Raul@gmail.com, ESmith@state. edu, DBrown@aol.com, jenny@aol.com, and R and y@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jenny to JKim@gmail.com. The second arrow goes from Jenny to jenny@aol.com. The third arrow goes from R and y to R and y@gmail.com. The fourth arrow goes from Dennis to DBrown@aol.com. The fifth arrow goes from Emily to ESmith@state. edu. The sixth arrow goes from Raul to RHern and ez@state. edu. The seventh arrow goes from Raul to Raul@gmail.com.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_211_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cJenny\u201d, \u201cR and y\u201d, \u201cDennis\u201d, \u201cEmily\u201d, and \u201cRaul\u201d. The table on the right has the header \u201cEmail\u201d and lists the email addresses RHern and ez@state. edu, JKim@gmail.com, Raul@gmail.com, ESmith@state. edu, DBrown@aol.com, jenny@aol.com, and R and y@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jenny to JKim@gmail.com. The second arrow goes from Jenny to jenny@aol.com. The third arrow goes from R and y to R and y@gmail.com. The fourth arrow goes from Dennis to DBrown@aol.com. The fifth arrow goes from Emily to ESmith@state. edu. The sixth arrow goes from Raul to RHern and ez@state. edu. The seventh arrow goes from Raul to Raul@gmail.com.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167824737653\"><p><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> {Jenny, R and y, Dennis, Emily, Raul} <span class=\"token\">\u24d2<\/span> {RHern and ez@state.edu, JKim@gmail.com, Raul@gmail.com, ESmith@state.edu, DBroen@aol.com, jenny@aol.cvom, R and y@gmail.com}<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833024527\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833256809\"><p id=\"fs-id1167833256811\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167833256812\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cJon\u201d, \u201cRachel\u201d, \u201cMatt\u201d, \u201cLeslie\u201d, \u201cChris\u201d, \u201cBeth\u201d, and \u201cLiz\u201d. The table on the right has the header \u201cEmail\u201d and lists the email addresses chrisg@gmail.com, lizzie@aol.com, jong@gmail.com, mattg@gmail.com, Rachel@state. edu, leslie@aol.com, and bethc@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jon to jong@gmail.com. The second arrow goes from Rachel to Rachel@state. edu. The third arrow goes from Matt to mattg@gmail.com. The fourth arrow goes from Leslie to leslie@aol.com. The fifth arrow goes from Chris to chrisg@gmail.com. The sixth arrow goes from Beth to bethc@gmail.com. The seventh arrow goes from Liz to lizzie@aol.com.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_212_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cJon\u201d, \u201cRachel\u201d, \u201cMatt\u201d, \u201cLeslie\u201d, \u201cChris\u201d, \u201cBeth\u201d, and \u201cLiz\u201d. The table on the right has the header \u201cEmail\u201d and lists the email addresses chrisg@gmail.com, lizzie@aol.com, jong@gmail.com, mattg@gmail.com, Rachel@state. edu, leslie@aol.com, and bethc@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jon to jong@gmail.com. The second arrow goes from Rachel to Rachel@state. edu. The third arrow goes from Matt to mattg@gmail.com. The fourth arrow goes from Leslie to leslie@aol.com. The fifth arrow goes from Chris to chrisg@gmail.com. The sixth arrow goes from Beth to bethc@gmail.com. The seventh arrow goes from Liz to lizzie@aol.com.\"><\/span><\/div><\/div><p id=\"fs-id1167836705602\">In the following exercises, determine whether each equation is a function.<\/p><div data-type=\"exercise\" id=\"fs-id1167836705606\"><div data-type=\"problem\" id=\"fs-id1167836525998\"><p id=\"fs-id1167836526000\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(2x+y=-3\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(y={x}^{2}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(x+{y}^{2}=-5\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836609321\"><p id=\"fs-id1167836650178\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> no<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836480280\"><div data-type=\"problem\" id=\"fs-id1167829828558\"><p id=\"fs-id1167829828560\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(y=3x-5\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(y={x}^{3}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(2x+{y}^{2}=4\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836526018\"><div data-type=\"problem\" id=\"fs-id1167836526020\"><p id=\"fs-id1167836526022\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(y-3{x}^{3}=2\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(x+{y}^{2}=3\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(3x-2y=6\\)<\/div><div data-type=\"solution\" id=\"fs-id1167833135088\"><p id=\"fs-id1167833135090\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no <span class=\"token\">\u24d2<\/span> yes<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833212947\"><div data-type=\"problem\" id=\"fs-id1167833212949\"><p id=\"fs-id1167836521932\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(2x-4y=8\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-4={x}^{2}-y\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({y}^{2}=\\text{\u2212}x+5\\)<\/div><\/div><p id=\"fs-id1167836714017\"><strong data-effect=\"bold\">Find the Value of a Function<\/strong><\/p><p id=\"fs-id1167829717540\">In the following exercises, evaluate the function: <span class=\"token\">\u24d0<\/span> \\(f\\left(2\\right)\\) <span class=\"token\">\u24d1<\/span> \\(f\\left(-1\\right)\\) <span class=\"token\">\u24d2<\/span> \\(f\\left(a\\right).\\)<\/p><div data-type=\"exercise\" id=\"fs-id1167823013283\"><div data-type=\"problem\" id=\"fs-id1167823013285\"><p id=\"fs-id1167823013287\">\\(f\\left(x\\right)=5x-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829624155\"><p id=\"fs-id1167829624157\"><span class=\"token\">\u24d0<\/span>\\(f\\left(2\\right)=7\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(-1\\right)=-8\\)<span class=\"token\">\u24d2<\/span>\\(f\\left(a\\right)=5a-3\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167829747304\">\\(f\\left(x\\right)=3x+4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829744293\"><div data-type=\"problem\" id=\"fs-id1167829744295\"><p id=\"fs-id1167836423791\">\\(f\\left(x\\right)=-4x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833047067\"><p id=\"fs-id1167836690281\"><span class=\"token\">\u24d0<\/span>\\(f\\left(2\\right)=-6\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(-1\\right)=6\\)<span class=\"token\">\u24d2<\/span>\\(f\\left(a\\right)=-4a+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836391029\"><div data-type=\"problem\" id=\"fs-id1167836391032\"><p id=\"fs-id1167836391034\">\\(f\\left(x\\right)=-6x-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829877909\"><div data-type=\"problem\" id=\"fs-id1167836575834\"><p id=\"fs-id1167836575836\">\\(f\\left(x\\right)={x}^{2}-x+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824737178\"><p id=\"fs-id1167836692744\"><span class=\"token\">\u24d0<\/span>\\(f\\left(2\\right)=5\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(-1\\right)=5\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(f\\left(a\\right)={a}^{2}-a+3\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836787728\"><div data-type=\"problem\" id=\"fs-id1167829789882\"><p id=\"fs-id1167829789884\">\\(f\\left(x\\right)={x}^{2}+x-2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836574347\"><div data-type=\"problem\" id=\"fs-id1167836574349\"><p id=\"fs-id1167829695807\">\\(f\\left(x\\right)=2{x}^{2}-x+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829689462\"><p id=\"fs-id1167829689464\"><span class=\"token\">\u24d0<\/span>\\(f\\left(2\\right)=9\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(-1\\right)=6\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(f\\left(a\\right)=2{a}^{2}-a+3\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829877536\"><div data-type=\"problem\" id=\"fs-id1167836791925\"><p id=\"fs-id1167836791928\">\\(f\\left(x\\right)=3{x}^{2}+x-2\\)<\/p><\/div><\/div><p id=\"fs-id1167836575175\">In the following exercises, evaluate the function: <span class=\"token\">\u24d0<\/span> \\(g\\left({h}^{2}\\right)\\) <span class=\"token\">\u24d1<\/span> \\(g\\left(x+2\\right)\\) <span class=\"token\">\u24d2<\/span> \\(g\\left(x\\right)+g\\left(2\\right).\\)<\/p><div data-type=\"exercise\" id=\"fs-id1167829595951\"><div data-type=\"problem\" id=\"fs-id1167829595953\"><p id=\"fs-id1167826132424\">\\(g\\left(x\\right)=2x+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836575320\"><p id=\"fs-id1167836575323\"><span class=\"token\">\u24d0<\/span>\\(g\\left({h}^{2}\\right)=2{h}^{2}+1\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(g\\left(x+2\\right)=4x+5\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(g\\left(x\\right)+g\\left(2\\right)=2x+6\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836399733\"><div data-type=\"problem\" id=\"fs-id1167836399735\"><p id=\"fs-id1167833208016\">\\(g\\left(x\\right)=5x-8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833054462\"><div data-type=\"problem\" id=\"fs-id1167833054464\"><p id=\"fs-id1167829651167\">\\(g\\left(x\\right)=-3x-2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829754538\"><p id=\"fs-id1167829754540\"><span class=\"token\">\u24d0<\/span>\\(g\\left({h}^{2}\\right)=-3{h}^{2}-2\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(g\\left(x+2\\right)=-3x-8\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(g\\left(x\\right)+g\\left(2\\right)=-3x-10\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829924597\"><div data-type=\"problem\" id=\"fs-id1167829924599\"><p id=\"fs-id1167829924601\">\\(g\\left(x\\right)=-8x+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836613442\"><div data-type=\"problem\" id=\"fs-id1167836613444\"><p id=\"fs-id1167836613446\">\\(g\\left(x\\right)=3-x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824976320\"><p id=\"fs-id1167836553636\"><span class=\"token\">\u24d0<\/span>\\(g\\left({h}^{2}\\right)=3-{h}^{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(g\\left(x+2\\right)=1-x\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(g\\left(x\\right)+g\\left(2\\right)=4-x\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833381469\"><div data-type=\"problem\" id=\"fs-id1167829860145\"><p id=\"fs-id1167829860148\">\\(g\\left(x\\right)=7-5x\\)<\/p><\/div><\/div><p id=\"fs-id1167826171384\">In the following exercises, evaluate the function.<\/p><div data-type=\"exercise\" id=\"fs-id1167826025131\"><div data-type=\"problem\" id=\"fs-id1167826025133\"><p id=\"fs-id1167826025135\">\\(f\\left(x\\right)=3{x}^{2}-5x;\\)\\(f\\left(2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829879609\"><p id=\"fs-id1167829879611\">2<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833240212\"><div data-type=\"problem\" id=\"fs-id1167833240214\"><p id=\"fs-id1167826206218\">\\(g\\left(x\\right)=4{x}^{2}-3x;\\)\\(g\\left(3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829713382\"><div data-type=\"problem\" id=\"fs-id1167829713384\"><p id=\"fs-id1167829713386\">\\(F\\left(x\\right)=2{x}^{2}-3x+1;\\)<\/p><div data-type=\"newline\"><br><\/div>\\(F\\left(-1\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167829651457\"><p id=\"fs-id1167829651459\">6<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833382456\"><div data-type=\"problem\" id=\"fs-id1167833382458\"><p id=\"fs-id1167833382460\">\\(G\\left(x\\right)=3{x}^{2}-5x+2;\\)<\/p><div data-type=\"newline\"><br><\/div>\\(G\\left(-2\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833052040\"><div data-type=\"problem\" id=\"fs-id1167833052042\"><p id=\"fs-id1167833052044\">\\(h\\left(t\\right)=2|t-5|+4;\\)\\(f\\left(-4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824976354\"><p id=\"fs-id1167824976356\">22<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836649081\"><div data-type=\"problem\" id=\"fs-id1167836649083\"><p id=\"fs-id1167836649085\">\\(h\\left(y\\right)=3|y-1|-3;\\)\\(h\\left(-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829906037\"><div data-type=\"problem\" id=\"fs-id1167829906040\"><p id=\"fs-id1167829906042\">\\(f\\left(x\\right)=\\frac{x+2}{x-1};\\)\\(f\\left(2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829579006\"><p id=\"fs-id1167829579008\">4<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833309035\"><div data-type=\"problem\" id=\"fs-id1167833309037\"><p id=\"fs-id1167833309039\">\\(g\\left(x\\right)=\\frac{x-2}{x+2};\\)\\(g\\left(4\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167836409828\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167836409832\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833386164\"><p id=\"fs-id1167833386167\">The number of unwatched shows in Sylvia\u2019s DVR is 85. This number grows by 20 unwatched shows per week. The function \\(N\\left(t\\right)=85+20t\\) represents the relation between the number of unwatched shows, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in weeks.<\/p><p id=\"fs-id1167824737629\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p><p id=\"fs-id1167836697913\"><span class=\"token\">\u24d1<\/span> Find \\(N\\left(4\\right).\\) Explain what this result means<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833378063\"><p id=\"fs-id1167832935494\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">t<\/em> IND; <em data-effect=\"italics\">N<\/em> DEP<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(N\\left(4\\right)=165\\) the number of unwatched shows in Sylvia\u2019s DVR at the fourth week. <\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167825884747\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167825884749\"><p id=\"fs-id1167825884751\">Every day a new puzzle is downloaded into Ken\u2019s account. Right now he has 43 puzzles in his account. The function \\(N\\left(t\\right)=43+t\\) represents the relation between the number of puzzles, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in days.<\/p><p id=\"fs-id1167829620296\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p><p id=\"fs-id1167829620303\"><span class=\"token\">\u24d1<\/span> Find \\(N\\left(30\\right).\\) Explain what this result means.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833380107\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836574149\"><p id=\"fs-id1167836574151\">The daily cost to the printing company to print a book is modeled by the function \\(C\\left(x\\right)=3.25x+1500\\) where <em data-effect=\"italics\">C<\/em> is the total daily cost and <em data-effect=\"italics\">x<\/em> is the number of books printed.<\/p><p id=\"fs-id1167833196806\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p><p id=\"fs-id1167826077595\"><span class=\"token\">\u24d1<\/span> Find \\(N\\left(0\\right).\\) Explain what this result means.<\/p><p id=\"fs-id1167829831994\"><span class=\"token\">\u24d2<\/span> Find \\(N\\left(1000\\right).\\) Explain what this result means.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836406944\"><p id=\"fs-id1167836406946\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">x<\/em> IND; <em data-effect=\"italics\">C<\/em> DEP<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(N\\left(0\\right)=1500\\) the daily cost if no books are printed<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> \\(N\\left(1000\\right)=4750\\) the daily cost of printing 1000 books<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829749356\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829749358\"><p id=\"fs-id1167829749361\">The daily cost to the manufacturing company is modeled by the function \\(C\\left(x\\right)=7.25x+2500\\) where \\(C\\left(x\\right)\\) is the total daily cost and <em data-effect=\"italics\">x<\/em> is the number of items manufactured.<\/p><p id=\"fs-id1167829879667\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p><p id=\"fs-id1167833272975\"><span class=\"token\">\u24d1<\/span> Find \\(C\\left(0\\right).\\) Explain what this result means.<\/p><p id=\"fs-id1167836494341\"><span class=\"token\">\u24d2<\/span> Find \\(C\\left(1000\\right).\\) Explain what this result means.<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167829756260\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167829756267\"><div data-type=\"problem\" id=\"fs-id1167829878963\"><p id=\"fs-id1167829878966\">In your own words, explain the difference between a relation and a function.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829878971\"><div data-type=\"problem\" id=\"fs-id1167833054106\"><p id=\"fs-id1167833054108\">In your own words, explain what is meant by domain and range.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833054114\"><div data-type=\"problem\" id=\"fs-id1167836755007\"><p id=\"fs-id1167836755009\">Is every relation a function? Is every function a relation?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836755014\"><div data-type=\"problem\" id=\"fs-id1167829718391\"><p id=\"fs-id1167829718394\">How do you find the value of a function?<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829783756\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167829783762\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167836607305\" data-alt=\"The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cconfidently\u201d, the third is \u201cwith some help\u201d, \u201cno minus I don\u2019t get it!\u201d. Under the first column are the phrases \u201cfind the domain and range of a relation\u201d, \u201cdetermine if a relation is a function\u201d, and \u201cfind the value of a function\u201d. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_213_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cconfidently\u201d, the third is \u201cwith some help\u201d, \u201cno minus I don\u2019t get it!\u201d. Under the first column are the phrases \u201cfind the domain and range of a relation\u201d, \u201cdetermine if a relation is a function\u201d, and \u201cfind the value of a function\u201d. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.\"><\/span><p id=\"fs-id1167836607297\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167829808856\"><dt>domain of a relation<\/dt><dd id=\"fs-id1167836729545\">The domain of a relation is all the <em data-effect=\"italics\">x<\/em>-values in the ordered pairs of the relation.<\/dd><\/dl><dl id=\"fs-id1167833327007\"><dt>function<\/dt><dd id=\"fs-id1167833327010\">A function is a relation that assigns to each element in its domain exactly one element in the range.<\/dd><\/dl><dl id=\"fs-id1167833327015\"><dt>mapping<\/dt><dd id=\"fs-id1167833382797\">A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range.<\/dd><\/dl><dl id=\"fs-id1167833382803\"><dt>range of a relation<\/dt><dd id=\"fs-id1167832978259\">The range of a relation is all the <em data-effect=\"italics\">y-<\/em>values in the ordered pairs of the relation.<\/dd><\/dl><dl id=\"fs-id1167833175472\"><dt>relation<\/dt><dd id=\"fs-id1167833175475\">A relation is any set of ordered pairs,\\(\\left(x,y\\right).\\) All the <em data-effect=\"italics\">x<\/em>-values in the ordered pairs together make up the domain. All the <em data-effect=\"italics\">y<\/em>-values in the ordered pairs together make up the range.<\/dd><\/dl><\/div>\n","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 the domain and range of a relation<\/li>\n<li>Determine if a relation is a function<\/li>\n<li>Find the value of a function<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836299681\" class=\"be-prepared\">\n<p id=\"fs-id1167829627915\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167836635622\" type=\"1\">\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0cde4cf16a0e0570f842e33209bc778_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/>.\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8740b4cf343dc102268a444dc1171244_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"89\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c2ae728b9a161e044125a0dc6c913c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"47\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" 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-6509e1ef9be32a6f74a385f1bad824ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#120;&#45;&#49;&#45;&#52;&#120;&#43;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"125\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836652573\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829789538\">\n<h3 data-type=\"title\">Find the Domain and Range of a Relation<\/h3>\n<p id=\"fs-id1167836538118\">As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number and our birthday are matched to our name. There is a relationship between our name and each of those items.<\/p>\n<p id=\"fs-id1167829921677\">When your professor gets her class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student\u2019s ID number, we call this a <span data-type=\"term\">relation<\/span>.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836418492\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27d5bc353a45ea1836682bd7f6864d7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#83;&#116;&#117;&#100;&#101;&#110;&#116;&#32;&#110;&#97;&#109;&#101;&#44;&#32;&#83;&#116;&#117;&#100;&#101;&#110;&#116;&#32;&#73;&#68;&#32;&#35;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"223\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167836547061\">The set of all the names of the students in the class is called the <span data-type=\"term\">domain<\/span> of the relation and the set of all student ID numbers paired with these students is the range of the relation.<\/p>\n<p id=\"fs-id1167833059636\">There are many similar situations where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836378970\">\n<div data-type=\"title\">Relation<\/div>\n<p id=\"fs-id1167824735594\">A <strong data-effect=\"bold\">relation<\/strong> is any set of ordered pairs,<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ab61ec2033ed5b69d0447eac5d6a4f8_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> All the <em data-effect=\"italics\">x<\/em>-values in the ordered pairs together make up the <strong data-effect=\"bold\">domain<\/strong>. All the <em data-effect=\"italics\">y<\/em>-values in the ordered pairs together make up the <strong data-effect=\"bold\">range<\/strong>.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836692527\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836327016\">\n<div data-type=\"problem\" id=\"fs-id1167833385954\">\n<p id=\"fs-id1167826194732\">For the relation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfe241bb51dbb4043afd846e1b5eb8b4_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;&#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;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"284\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167836619755\"><span class=\"token\">\u24d0<\/span> Find the domain of the relation.<\/p>\n<p id=\"fs-id1167833356505\"><span class=\"token\">\u24d1<\/span> Find the range of the relation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824765140\">\n<p id=\"fs-id1167836597346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e281fa0404b51a5ad6186ba5c8151a50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#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;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"274\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167829691118\"><span class=\"token\">\u24d0<\/span> The domain is the set of all <em data-effect=\"italics\">x<\/em>-values of the relation. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e41c38859d91f2989b5f200f5f5e62af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#49;&#44;&#50;&#44;&#51;&#44;&#52;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167829593929\"><span class=\"token\">\u24d1<\/span> The range is the set of all <em data-effect=\"italics\">y<\/em>-values of the relation. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06ca12cade89f05a33c9de54b600586d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#49;&#44;&#52;&#44;&#57;&#44;&#49;&#54;&#44;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836629801\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833272898\">\n<div data-type=\"problem\" id=\"fs-id1167826169765\">\n<p>For the relation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81395695549e5121b2388177b7a32df8_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;&#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;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#54;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#49;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"301\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167836550500\"><span class=\"token\">\u24d0<\/span> Find the domain of the relation.<\/p>\n<p id=\"fs-id1167833202428\"><span class=\"token\">\u24d1<\/span> Find the range of the relation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836618853\">\n<p id=\"fs-id1167836492201\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80baeb391d54c83c19be08d8f7a35e49_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;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" 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-555d070441ef28721b0731bf4051e2c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#49;&#44;&#56;&#44;&#50;&#55;&#44;&#54;&#52;&#44;&#49;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833021555\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829614386\">\n<p id=\"fs-id1167836360520\">For the relation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00397d80d4f7db779f76ef18427d27f5_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;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"284\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167833290815\"><span class=\"token\">\u24d0<\/span> Find the domain of the relation.<\/p>\n<p id=\"fs-id1167825702514\"><span class=\"token\">\u24d1<\/span> Find the range of the relation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829812976\">\n<p id=\"fs-id1167833346148\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80baeb391d54c83c19be08d8f7a35e49_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;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" 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-6e5def6c2cbfc9a51bd658b23caec01a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#51;&#44;&#54;&#44;&#57;&#44;&#49;&#50;&#44;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836509402\">\n<div data-type=\"title\">Mapping<\/div>\n<p id=\"fs-id1167824927876\">A <span data-type=\"term\">mapping<\/span> is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167829683746\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836525923\">\n<p id=\"fs-id1167833412571\">Use the <strong data-effect=\"bold\">mapping<\/strong> of the relation shown to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation, <span class=\"token\">\u24d1<\/span> find the domain of the relation, and <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833364760\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cAlison\u201d, \u201cPenelope\u201d, \u201cJune\u201d, \u201cGregory\u201d, \u201cGeoffrey\u201d, \u201cLauren\u201d, \u201cStephen\u201d, \u201cAlice\u201d, \u201cLiz\u201d, \u201cDanny\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 12\u201d, \u201cFebruary 3\u201d, \u201cApril 25\u201d, \u201cMay 10\u201d, \u201cMay 23\u201d, \u201cJuly 24\u201d, \u201cAugust 2\u201d, and \u201cSeptember 15\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a date in the Birthday table. While most dates have only one arrow pointing to them, there are two arrows pointing to July 24: one from Stephen and one from Liz.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cAlison\u201d, \u201cPenelope\u201d, \u201cJune\u201d, \u201cGregory\u201d, \u201cGeoffrey\u201d, \u201cLauren\u201d, \u201cStephen\u201d, \u201cAlice\u201d, \u201cLiz\u201d, \u201cDanny\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 12\u201d, \u201cFebruary 3\u201d, \u201cApril 25\u201d, \u201cMay 10\u201d, \u201cMay 23\u201d, \u201cJuly 24\u201d, \u201cAugust 2\u201d, and \u201cSeptember 15\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a date in the Birthday table. While most dates have only one arrow pointing to them, there are two arrows pointing to July 24: one from Stephen and one from Liz.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836417183\">\n<p id=\"fs-id1167836327324\"><span class=\"token\">\u24d0<\/span> The arrow shows the matching of the person to their birthday. We create ordered pairs with the person\u2019s name as the <em data-effect=\"italics\">x<\/em>-value and their birthday as the <em data-effect=\"italics\">y<\/em>-value.<\/p>\n<p id=\"fs-id1167833008731\">{(Alison, April 25), (Penelope, May 23), (June, August 2), (Gregory, September 15), (Geoffrey, January 12), (Lauren, May 10), (Stephen, July 24), (Alice, February 3), (Liz, August 2), (Danny, July 24)}<\/p>\n<p id=\"fs-id1167836788581\"><span class=\"token\">\u24d1<\/span> The domain is the set of all <em data-effect=\"italics\">x<\/em>-values of the relation.<\/p>\n<p id=\"fs-id1167836648651\">{Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, Danny}<\/p>\n<p id=\"fs-id1167836689824\"><span class=\"token\">\u24d2<\/span> The range is the set of all <em data-effect=\"italics\">y<\/em>-values of the relation.<\/p>\n<p id=\"fs-id1167836326011\">{January 12, February 3, April 25, May 10, May 23, July 24, August 2, September 15}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836340432\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829714243\">\n<p id=\"fs-id1167829614511\">Use the mapping of the relation shown to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<p><span data-type=\"media\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cKhanh Nguyen\u201d, \u201cAbigail Brown\u201d, \u201cSumantha Mishal\u201d, and \u201cJose Hern and ez\u201d. The table on the right has the header \u201cStudent ID #\u201d and lists the codes \u201ca b 56781\u201d, \u201cj h 47983\u201d, \u201ck n 68413\u201d, and \u201cs m 32479\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a code in the student ID table. The first arrow goes from Khanh Nguyen to k n 68413. The second arrow goes from Abigail Brown to a b 56781. The third arrow goes from Sumantha Mishal to s m 32479. The fourth arrow goes from Jose Hern and ez to j h 47983.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cKhanh Nguyen\u201d, \u201cAbigail Brown\u201d, \u201cSumantha Mishal\u201d, and \u201cJose Hern and ez\u201d. The table on the right has the header \u201cStudent ID #\u201d and lists the codes \u201ca b 56781\u201d, \u201cj h 47983\u201d, \u201ck n 68413\u201d, and \u201cs m 32479\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a code in the student ID table. The first arrow goes from Khanh Nguyen to k n 68413. The second arrow goes from Abigail Brown to a b 56781. The third arrow goes from Sumantha Mishal to s m 32479. The fourth arrow goes from Jose Hern and ez to j h 47983.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836554322\">\n<p id=\"fs-id1167836539147\"><span class=\"token\">\u24d0<\/span> (Khanh Nguyen, kn68413), (Abigail Brown, ab56781), (Sumantha Mishal, sm32479), (Jose Hern and ez, jh47983) <span class=\"token\">\u24d1<\/span> {Khanh Nguyen, Abigail Brown, Sumantha Mishal, Jose Hern and ez} <span class=\"token\">\u24d2<\/span> {kn68413, ab56781, sm32479, jh47983}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824754892\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836663646\">\n<div data-type=\"problem\" id=\"fs-id1167829716810\">\n<p id=\"fs-id1167836731398\">Use the mapping of the relation shown to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833385478\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cMaria\u201d, \u201cArm and o\u201d, \u201cCynthia\u201d, \u201cKelly\u201d, and \u201cRachel\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 18\u201d, \u201cMarch 15\u201d, \u201cNovember 6\u201d, and \u201cDecember 8\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a date in the Birthday table. The first arrow goes from Maria to November 6. The second arrow goes from Arm and o to a January 18. The third arrow goes from Cynthia to December 8. The fourth arrow goes from Kelly to March 15. The fifth arrow goes from Rachel to November 6.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cMaria\u201d, \u201cArm and o\u201d, \u201cCynthia\u201d, \u201cKelly\u201d, and \u201cRachel\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 18\u201d, \u201cMarch 15\u201d, \u201cNovember 6\u201d, and \u201cDecember 8\u201d. There is one arrow for each name in the Name table that starts at the name and points toward a date in the Birthday table. The first arrow goes from Maria to November 6. The second arrow goes from Arm and o to a January 18. The third arrow goes from Cynthia to December 8. The fourth arrow goes from Kelly to March 15. The fifth arrow goes from Rachel to November 6.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750633\">\n<p><span class=\"token\">\u24d0<\/span> (Maria, November 6), (Arm and o, January 18), (Cynthia, December 8), (Kelly, March 15), (Rachel, November 6) <span class=\"token\">\u24d1<\/span> {Maria, Arm and o, Cynthia, Kelly, Rachel} <span class=\"token\">\u24d2<\/span> {November 6, January 18, December 8, March 15}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836532042\">A graph is yet another way that a relation can be represented. The set of ordered pairs of all the points plotted is the relation. The set of all <em data-effect=\"italics\">x<\/em>-coordinates is the domain of the relation and the set of all <em data-effect=\"italics\">y<\/em>-coordinates is the range. Generally we write the numbers in ascending order for both the domain and range.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833057329\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829877976\">\n<div data-type=\"problem\" id=\"fs-id1167836450184\">\n<p id=\"fs-id1167829681249\">Use the graph of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836673420\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 1), (0, 3), (1, 5), (2, negative 2), and (4, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 1), (0, 3), (1, 5), (2, negative 2), and (4, negative 2).\" \/><\/span><\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836613766\"><span class=\"token\">\u24d0<\/span> The ordered pairs of the relation are: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3de35737ee24a452db2b2d30a63e4381_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#53;&#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;&#52;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#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=\"383\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167836429436\"><span class=\"token\">\u24d1<\/span> The domain is the set of all <em data-effect=\"italics\">x<\/em>-values of the relation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63203ebe9c827ebbec56423054033021_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#51;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167833137637\">Notice that while <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> repeats, it is only listed once.<\/p>\n<p id=\"fs-id1167833364739\"><span class=\"token\">\u24d2<\/span> The range is the set of all <em data-effect=\"italics\">y<\/em>-values of the relation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bfd33225df5cc260508927b8ff6e768_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#50;&#44;&#45;&#49;&#44;&#51;&#44;&#52;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167833050654\">Notice that while <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> repeats, it is only listed once.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829750323\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829850985\">\n<div data-type=\"problem\" id=\"fs-id1167829879283\">\n<p id=\"fs-id1167824773962\">Use the graph of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836341066\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 3), (negative 2, 2), (negative 1, 0), (0, negative 1), (2, negative 2), and (4, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 3), (negative 2, 2), (negative 1, 0), (0, negative 1), (2, negative 2), and (4, negative 4).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836419079\">\n<p id=\"fs-id1167833051815\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-558e7d273d31ee3e093c5149b1087b53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#51;&#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;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"187\" style=\"vertical-align: -4px;\" \/><\/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-33862af0b60d6769dc609466d4b7f874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"180\" 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-6b12f971aedfc20e6ba70be9cda76251_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -5px;\" \/><\/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-c664fbc92dd09c863b74d9007b1cc88c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#51;&#44;&#50;&#44;&#48;&#44;&#45;&#49;&#44;&#45;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829738598\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833056928\">\n<div data-type=\"problem\" id=\"fs-id1167836485775\">\n<p id=\"fs-id1167833050002\">Use the graph of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833049956\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 5), (negative 3, 0), (negative 3, negative 6), (negative 1, negative 2), (1, 2), and (4, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 5), (negative 3, 0), (negative 3, negative 6), (negative 1, negative 2), (1, 2), and (4, negative 4).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833021407\">\n<p id=\"fs-id1167836503993\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-385938c1de185ec23af2729806a73d50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#51;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"201\" style=\"vertical-align: -4px;\" \/><\/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-620d7d047b8b7a070fe69a0471f6983f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"180\" 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-6ed576fdd264765977f40423742f5c12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#51;&#44;&#45;&#49;&#44;&#49;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -5px;\" \/><\/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-6dbaf842b87bad5307cd6f987744539a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#54;&#44;&#48;&#44;&#53;&#44;&#45;&#50;&#44;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836610583\">\n<h3 data-type=\"title\">Determine if a Relation is a Function<\/h3>\n<p id=\"fs-id1167826171759\">A special type of relation, called a <span data-type=\"term\">function<\/span>, occurs extensively in mathematics. A function 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<div data-type=\"note\" id=\"fs-id1167836614987\">\n<div data-type=\"title\">Function<\/div>\n<p id=\"fs-id1167829714644\">A <strong data-effect=\"bold\">function<\/strong> is a relation that assigns to each element in its domain exactly one element in the range.<\/p>\n<\/div>\n<p id=\"fs-id1167836532939\">The birthday example from <a href=\"#fs-id1167829683746\" class=\"autogenerated-content\">(Figure)<\/a> helps us understand this definition. Every person has a birthday but no one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24<sup>th<\/sup> as their birthday and that June and Liz share August 2<sup>nd<\/sup>. Since each person has exactly one birthday, the relation in <a href=\"#fs-id1167829683746\" class=\"autogenerated-content\">(Figure)<\/a> is a function.<\/p>\n<p id=\"fs-id1167836601426\">The relation shown by the graph in <a href=\"#fs-id1167833057329\" class=\"autogenerated-content\">(Figure)<\/a> includes the ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d29d505cec09336aa569e1cca8670699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9edb132d6e0d9a8e7cebf538ef2f84a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> Is that okay in a function? No, as this is like one person having two different birthdays.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167824733092\">\n<div data-type=\"problem\" id=\"fs-id1167826132575\">\n<p id=\"fs-id1167836456488\">Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the relation.<\/p>\n<p id=\"fs-id1167836518420\"><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;\" \/><\/p>\n<p id=\"fs-id1167824735617\"><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70b3912ab3d34a0c55485f18e3dc2e81_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;&#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;&#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-id1167833052137\">\n<p id=\"fs-id1167836511265\"><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;\" \/><\/p>\n<p id=\"fs-id1167836691895\">(i) 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>(ii) The domain is the set of all <em data-effect=\"italics\">x<\/em>-values in the relation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The domain is: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-010457c6e14cdf763f9a4730f613b7ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"174\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167836282463\">(iii) The range is the set of all <em data-effect=\"italics\">y<\/em>-values in the relation. Notice we do not list range values twice.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The range is: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0b9d5ecbbe302fad93c73cd8cbf13cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#50;&#55;&#44;&#56;&#44;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167824735962\"><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70b3912ab3d34a0c55485f18e3dc2e81_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;&#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;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"398\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167833223866\">(i) The <em data-effect=\"italics\">x<\/em>-value 9 is matched with two <em data-effect=\"italics\">y<\/em>-values, both 3 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfa36400e576c82fd4847d2da37b1d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> So this relation is not a function.<\/p>\n<p>(ii) The domain is the set of all <em data-effect=\"italics\">x<\/em>-values in the relation. Notice we do not list domain values twice.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The domain is: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff9c16494ea857b82457a5c0c00f6714_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;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -5px;\" \/><\/p>\n<p id=\"fs-id1167836507848\">(iii) The range is the set of all <em data-effect=\"italics\">y<\/em>-values in the relation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The range is: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-010457c6e14cdf763f9a4730f613b7ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"174\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829742590\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836544009\">\n<p id=\"fs-id1167833060073\">Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the function.<\/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-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<p id=\"fs-id1167833380236\"><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73f73a38f8c7275986091cc17aa207b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#45;&#52;&#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;&#50;&#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;&#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;&#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=\"398\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836352889\">\n<p id=\"fs-id1167826188989\"><span class=\"token\">\u24d0<\/span> Yes; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-549784072575e0f88f0accf97a0c3960_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"174\" 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-3e861f72d10223285e0088303ee6b499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#54;&#44;&#45;&#52;&#44;&#45;&#50;&#44;&#48;&#44;&#50;&#44;&#52;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"167\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> No; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a64633d8e94ede46cf3b90e26bfdb33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#48;&#44;&#50;&#44;&#52;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" 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-ed14b26ee91f48d7bb804db08a4710cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#52;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"167\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836362682\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836533244\">\n<div data-type=\"problem\" id=\"fs-id1167836717454\">\n<p id=\"fs-id1167836550940\">Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the relation.<\/p>\n<p id=\"fs-id1167836598862\"><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<p id=\"fs-id1167836310540\"><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3f043317ee4b5512d25b2e9d0a07bdf_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;&#45;&#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=\"453\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836298199\">\n<p id=\"fs-id1167836798080\"><span class=\"token\">\u24d0<\/span> No; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e82e6d25437730f0e8b8d9e842a880e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#48;&#44;&#49;&#44;&#56;&#44;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" 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-ffbd025c09fee52fbe0a57cfee729505_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#50;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"167\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Yes; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-665fe234825bf0e8f7ad503bd1742ed8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#55;&#44;&#45;&#53;&#44;&#56;&#44;&#48;&#44;&#45;&#54;&#44;&#45;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"188\" 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-3bb26efe5b4191b4493d38a71bfb57bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#45;&#51;&#44;&#45;&#52;&#44;&#48;&#44;&#52;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836539582\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829586801\">\n<div data-type=\"problem\" id=\"fs-id1167836684155\">\n<p id=\"fs-id1167833059664\">Use the mapping to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829942558\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cLydia\u201d, \u201cEugene\u201d, \u201cJanet\u201d, \u201cRick\u201d, and \u201cMarty\u201d. The table on the right has the header \u201cPhone number\u201d and lists the numbers \u201c321-549-3327 home\u201d, \u201c427-658-2314 cell\u201d, \u201c321-964-7324 cell\u201d, \u201c684-358-7961 home\u201d, \u201c684-369-7231 cell\u201d, and \u201c798-367-8541 cell\u201d. There are arrows that start at a name and points toward a number in the phone number table. The first arrow goes from Lydia to 321-549-3327 home. The second arrow goes from Lydia to a 321-964-7324 cell. The third arrow goes from Eugene to 427-658-2314 cell. The fourth arrow goes from Janet to 427-658-2314 cell. The fifth arrow goes from Rick to 798-367-8541 cell. The sixth arrow goes from Marty to 684-358-7961 home. The seventh arrow goes from Marty to 684-369-7231 cell.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cLydia\u201d, \u201cEugene\u201d, \u201cJanet\u201d, \u201cRick\u201d, and \u201cMarty\u201d. The table on the right has the header \u201cPhone number\u201d and lists the numbers \u201c321-549-3327 home\u201d, \u201c427-658-2314 cell\u201d, \u201c321-964-7324 cell\u201d, \u201c684-358-7961 home\u201d, \u201c684-369-7231 cell\u201d, and \u201c798-367-8541 cell\u201d. There are arrows that start at a name and points toward a number in the phone number table. The first arrow goes from Lydia to 321-549-3327 home. The second arrow goes from Lydia to a 321-964-7324 cell. The third arrow goes from Eugene to 427-658-2314 cell. The fourth arrow goes from Janet to 427-658-2314 cell. The fifth arrow goes from Rick to 798-367-8541 cell. The sixth arrow goes from Marty to 684-358-7961 home. The seventh arrow goes from Marty to 684-369-7231 cell.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836698471\">\n<p><span class=\"token\">\u24d0<\/span> Both Lydia and Marty have two phone numbers. So each <em data-effect=\"italics\">x<\/em>-value is not matched with only one <em data-effect=\"italics\">y<\/em>-value. So this relation is not a function.<\/p>\n<p id=\"fs-id1167833369757\"><span class=\"token\">\u24d1<\/span> The domain is the set of all <em data-effect=\"italics\">x<\/em>-values in the relation. The domain is: {Lydia, Eugene, Janet, Rick, Marty}<\/p>\n<p id=\"fs-id1167830076911\"><span class=\"token\">\u24d2<\/span> The range is the set of all <em data-effect=\"italics\">y<\/em>-values in the relation. The range is:<\/p>\n<p id=\"fs-id1167836576219\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b679b6db4dc0557c3f63fcb50467d8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#51;&#50;&#49;&#45;&#53;&#52;&#57;&#45;&#51;&#51;&#50;&#55;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -5px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1aed26563b6e5b3f13e84caa4001906_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#50;&#55;&#45;&#54;&#53;&#56;&#45;&#50;&#51;&#49;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d263eb236be3f329b61f86a19de0c28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#50;&#49;&#45;&#57;&#54;&#52;&#45;&#55;&#51;&#50;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b9bbcdd5c996f464c8503f58f1e113c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#56;&#52;&#45;&#51;&#53;&#56;&#45;&#55;&#57;&#54;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9cb2234bb2b18e51f4128737d973f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#56;&#52;&#45;&#51;&#54;&#57;&#45;&#55;&#50;&#51;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ea4c55ca920b6a3593da6c06772e4a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#57;&#56;&#45;&#51;&#54;&#55;&#45;&#56;&#53;&#52;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"132\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836378848\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833397170\">\n<div data-type=\"problem\" id=\"fs-id1167836361517\">\n<p>Use the mapping to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836392154\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNetwork\u201d and lists the television stations \u201cNBC\u201d, \u201cHGTV\u201d, and \u201cHBO\u201d. The table on the right has the header \u201cProgram\u201d and lists the television shows \u201cEllen Degeneres Show\u201d, \u201cLaw and Order\u201d, \u201cTonight Show\u201d, \u201cProperty Brothers\u201d, \u201cHouse Hunters\u201d, \u201cLove it or List it\u201d, \u201cGame of Thrones\u201d, \u201cTrue Detective\u201d, and \u201cSesame Street\u201d. There are arrows that start at a network in the first table and point toward a program in the second table. The first arrow goes from NBC to Ellen Degeneres Show. The second arrow goes from NBC to Law and Order. The third arrow goes from NBC to Tonight Show. The fourth arrow goes from HGTV to Property Brothers. The fifth arrow goes from HGTV to House Hunters. The sixth arrow goes from HGTV to Love it or List it. The seventh arrow goes from HBO to Game of Thrones. The eighth arrow goes from HBO to True Detective. The ninth arrow goes from HBO to Sesame Street.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNetwork\u201d and lists the television stations \u201cNBC\u201d, \u201cHGTV\u201d, and \u201cHBO\u201d. The table on the right has the header \u201cProgram\u201d and lists the television shows \u201cEllen Degeneres Show\u201d, \u201cLaw and Order\u201d, \u201cTonight Show\u201d, \u201cProperty Brothers\u201d, \u201cHouse Hunters\u201d, \u201cLove it or List it\u201d, \u201cGame of Thrones\u201d, \u201cTrue Detective\u201d, and \u201cSesame Street\u201d. There are arrows that start at a network in the first table and point toward a program in the second table. The first arrow goes from NBC to Ellen Degeneres Show. The second arrow goes from NBC to Law and Order. The third arrow goes from NBC to Tonight Show. The fourth arrow goes from HGTV to Property Brothers. The fifth arrow goes from HGTV to House Hunters. The sixth arrow goes from HGTV to Love it or List it. The seventh arrow goes from HBO to Game of Thrones. The eighth arrow goes from HBO to True Detective. The ninth arrow goes from HBO to Sesame Street.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836388771\">\n<p id=\"fs-id1167829830534\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> {NBC, HGTV, HBO} <span class=\"token\">\u24d2<\/span> {Ellen Degeneres Show, Law and Order, Tonight Show, Property Brothers, House Hunters, Love it or List it, Game of Thrones, True Detective, Sesame Street}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836511080\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836606033\">\n<div data-type=\"problem\" id=\"fs-id1167829579633\">\n<p id=\"fs-id1167833386907\">Use the mapping to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<p><span data-type=\"media\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cNeal\u201d, \u201cKrystal\u201d, \u201cKelvin\u201d, \u201cGeorge\u201d, \u201cChrista\u201d, and \u201cMike\u201d. The table on the right has the header \u201cPhone number\u201d and lists the numbers \u201c123-567-4389 work\u201d, \u201c231-378-5941 cell\u201d, \u201c753-469-9731 cell\u201d, \u201c567-534-2970 work\u201d, \u201c684-369-7231 cell\u201d, \u201c798-367-8541 cell\u201d, and \u201c639-847-6971 cell\u201d. There are arrows that start at a name and points toward a number in the phone number table. The first arrow goes from Neal to 753-469-9731 cell. The second arrow goes from Krystal to a 684-369-7231 cell. The third arrow goes from Kelvin to 231-378-5941 cell. The fourth arrow goes from George to 123-567-4389 work. The fifth arrow goes from George to 639-847-6971 cell. The sixth arrow goes from Christa to 567-534-2970 work. The seventh arrow goes from Mike to 567-534-2970 work. The eighth arrow goes from Mike to 798-367-8541 cell.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cNeal\u201d, \u201cKrystal\u201d, \u201cKelvin\u201d, \u201cGeorge\u201d, \u201cChrista\u201d, and \u201cMike\u201d. The table on the right has the header \u201cPhone number\u201d and lists the numbers \u201c123-567-4389 work\u201d, \u201c231-378-5941 cell\u201d, \u201c753-469-9731 cell\u201d, \u201c567-534-2970 work\u201d, \u201c684-369-7231 cell\u201d, \u201c798-367-8541 cell\u201d, and \u201c639-847-6971 cell\u201d. There are arrows that start at a name and points toward a number in the phone number table. The first arrow goes from Neal to 753-469-9731 cell. The second arrow goes from Krystal to a 684-369-7231 cell. The third arrow goes from Kelvin to 231-378-5941 cell. The fourth arrow goes from George to 123-567-4389 work. The fifth arrow goes from George to 639-847-6971 cell. The sixth arrow goes from Christa to 567-534-2970 work. The seventh arrow goes from Mike to 567-534-2970 work. The eighth arrow goes from Mike to 798-367-8541 cell.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836754813\">\n<p id=\"fs-id1167836407181\"><span class=\"token\">\u24d0<\/span> No <span class=\"token\">\u24d1<\/span> {Neal, Krystal, Kelvin, George, Christa, Mike} <span class=\"token\">\u24d2<\/span> {123-567-4839 work, 231-378-5941 cell, 743-469-9731 cell, 567-534-2970 work, 684-369-7231 cell, 798-367-8541 cell, 639-847-6971 cell}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836568027\">In algebra, more often than not, functions will be represented by an equation. It is easiest to see if the equation is a function when it is solved for <em data-effect=\"italics\">y<\/em>. If each value of <em data-effect=\"italics\">x<\/em> results in only one value of <em data-effect=\"italics\">y<\/em>, then the equation defines a function.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836683566\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836561265\">\n<div data-type=\"problem\" id=\"fs-id1167836506912\">\n<p id=\"fs-id1167829681109\">Determine whether each equation is a function.<\/p>\n<p id=\"fs-id1167833303503\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6eee09e6757335b61d3135eead59dc2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#121;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" 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-638e18d8d7b540a7734cd8d36769afea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" 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-f546a2df09d32fb7ec25f52dfe9f13fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836398976\">\n<p id=\"fs-id1167829599793\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6eee09e6757335b61d3135eead59dc2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#121;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167832926075\">For each value of <em data-effect=\"italics\">x<\/em>, we multiply it by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> and then add 7 to get the <em data-effect=\"italics\">y<\/em>-value<\/p>\n<table class=\"unnumbered unstyled\" summary=\"y = negative 2 x plus 7. For example, if x = 3. y = negative 2 times 3 plus 7. y = 1.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836495341\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">For example, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6bddd0431eca855e22d3dd4aa153594_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#58;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836619694\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836326343\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167833056514\">We have that when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7d62a78a97a33b2bfba1ddcfe9dbb8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4acfdc8013815a0ea0c7059786c6402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> It would work similarly for any value of <em data-effect=\"italics\">x<\/em>. Since each value of <em data-effect=\"italics\">x<\/em>, corresponds to only one value of <em data-effect=\"italics\">y<\/em> the equation defines a function.<\/p>\n<p id=\"fs-id1167836447354\"><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-638e18d8d7b540a7734cd8d36769afea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167829712213\">For each value of <em data-effect=\"italics\">x<\/em>, we square it and then add 1 to get the <em data-effect=\"italics\">y<\/em>-value.<\/p>\n<table id=\"fs-id1167836533787\" class=\"unnumbered unstyled\" summary=\"y = x squared plus 1. For example, if x = 2. y = 2 squared plus 1. y = 5.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547666\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">For example, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a70b7d0e1bce7596be64a282e1d39d7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#58;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836616301\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/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_03_05_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167829936815\">We have that when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e211b3d19a6898a4c9192f117c1fe08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf6f0342bc735be628e5e6062a1a2a19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> It would work similarly for any value of <em data-effect=\"italics\">x<\/em>. Since each value of <em data-effect=\"italics\">x<\/em>, corresponds to only one value of <em data-effect=\"italics\">y<\/em> the equation defines a function.<\/p>\n<p id=\"fs-id1167836515845\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829596595\" class=\"unnumbered unstyled\" summary=\"x plus y squared = 3. Isolate the y term. y squared = negative x plus 3. Let\u2019s substitute x = 2. y squared = negative 2 plus 3. y squared = 1.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829789899\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Isolate the <em data-effect=\"italics\">y<\/em> term.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833356461\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Let\u2019s substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d11400eca30b4e5ab418bd61ede75fa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836618915\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836296194\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>This give us two values for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67c921a8a8fbb42385af2441fc4a629c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#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;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836539656\">We have shown that when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e211b3d19a6898a4c9192f117c1fe08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e3ca3f6eb8810e090b4ceee7f6e129b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1144e182f26c1fd5166b5411a3ed3cd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/> It would work similarly for any value of <em data-effect=\"italics\">x<\/em>. Since each value of <em data-effect=\"italics\">x<\/em> does not corresponds to only one value of <em data-effect=\"italics\">y<\/em> the equation does not define a function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833379684\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836585298\">\n<div data-type=\"problem\" id=\"fs-id1167836356048\">\n<p id=\"fs-id1167829716494\">Determine whether each equation is a function.<\/p>\n<p id=\"fs-id1167836713610\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2457033f45011cf4dc8032671661a955_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#121;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" 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-a09ae22e07356fd47cbf537e2b9b9e78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" 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-f47adbfb448516055cd25cf89de143a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829811714\">\n<p id=\"fs-id1167833255910\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no <span class=\"token\">\u24d2<\/span> yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833369174\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836614714\">\n<p id=\"fs-id1167836730830\">Determine whether each equation is a function.<\/p>\n<p id=\"fs-id1167836621950\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9b87eb6bcd91f64b236b7b6b83a487c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" 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-cf33150a0ff8852dc0844dfe0b604732_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" 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-8156e618e133a329f5490c1325c25820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836289689\">\n<p id=\"fs-id1167829859295\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167824731607\">\n<h3 data-type=\"title\">Find the Value of a Function<\/h3>\n<p id=\"fs-id1167836315013\">It is very convenient to name a function and most often we name it <em data-effect=\"italics\">f<\/em>, <em data-effect=\"italics\">g<\/em>, <em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">F<\/em>, <em data-effect=\"italics\">G<\/em>, or <em data-effect=\"italics\">H<\/em>. In any function, for each <em data-effect=\"italics\">x<\/em>-value from the domain we get a corresponding <em data-effect=\"italics\">y<\/em>-value in the range. For the function <em data-effect=\"italics\">f<\/em>, we write this range value <em data-effect=\"italics\">y<\/em> as <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;\" \/> This is called function notation and is read <em data-effect=\"italics\">f<\/em> of <em data-effect=\"italics\">x<\/em> or the value of <em data-effect=\"italics\">f<\/em> at <em data-effect=\"italics\">x<\/em>. In this case the parentheses does not indicate multiplication.<\/p>\n<div data-type=\"note\" id=\"fs-id1167824735502\">\n<div data-type=\"title\">Function Notation<\/div>\n<p id=\"fs-id1167836379405\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70ecad17068db077c0b28295e4626839_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167829590438\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-019f1794939f117e00863a3e0352d17c_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;&#125;&#102;&#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;&#115;&#32;&#116;&#104;&#101;&#32;&#110;&#97;&#109;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#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;&#115;&#32;&#116;&#104;&#101;&#32;&#100;&#111;&#109;&#97;&#105;&#110;&#32;&#118;&#97;&#108;&#117;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#102;&#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;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#114;&#97;&#110;&#103;&#101;&#32;&#118;&#97;&#108;&#117;&#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;&#99;&#111;&#114;&#114;&#101;&#115;&#112;&#111;&#110;&#100;&#105;&#110;&#103;&#32;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#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;&#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=\"61\" width=\"417\" style=\"vertical-align: -26px;\" \/><\/div>\n<p>We read <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;\" \/> as <em data-effect=\"italics\">f<\/em> of <em data-effect=\"italics\">x<\/em> or the value of <em data-effect=\"italics\">f<\/em> at <em data-effect=\"italics\">x<\/em>.<\/p>\n<\/div>\n<p id=\"fs-id1167833186483\">We call <em data-effect=\"italics\">x<\/em> the independent variable as it can be any value in the domain. We call <em data-effect=\"italics\">y<\/em> the dependent variable as its value depends on <em data-effect=\"italics\">x<\/em>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829620795\">\n<div data-type=\"title\">Independent and Dependent Variables<\/div>\n<p id=\"fs-id1167836548512\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-220367de6dc72b963d97846ff4643d95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167836732021\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7d0a91d51740679fed02fa852db080e_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;&#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;&#115;&#32;&#116;&#104;&#101;&#32;&#105;&#110;&#100;&#101;&#112;&#101;&#110;&#100;&#101;&#110;&#116;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#32;&#97;&#115;&#32;&#105;&#116;&#32;&#99;&#97;&#110;&#32;&#98;&#101;&#32;&#97;&#110;&#121;&#32;&#118;&#97;&#108;&#117;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#100;&#111;&#109;&#97;&#105;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#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;&#116;&#104;&#101;&#32;&#100;&#101;&#112;&#101;&#110;&#100;&#101;&#110;&#116;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#32;&#97;&#115;&#32;&#105;&#116;&#115;&#32;&#118;&#97;&#108;&#117;&#101;&#32;&#100;&#101;&#112;&#101;&#110;&#100;&#115;&#32;&#111;&#110;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"514\" style=\"vertical-align: -15px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167836362234\">Much as when you first encountered the variable <em data-effect=\"italics\">x<\/em>, function notation may be rather unsettling. It seems strange because it is new. You will feel more comfortable with the notation as you use it.<\/p>\n<p id=\"fs-id1167833051398\">Let\u2019s look at the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6526a05cbb0a6fe2166e6970816f9cac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#120;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/> To find the value of <em data-effect=\"italics\">y<\/em> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e211b3d19a6898a4c9192f117c1fe08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> we know to substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> into the equation and then simplify.<\/p>\n<table class=\"unnumbered unstyled\" summary=\"y = 4 x minus 5. Let x = 2. y = 4 times 2 minus 5. y = 3.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836353205\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d11400eca30b4e5ab418bd61ede75fa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836433912\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832926879\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167829718470\">The value of the function at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> is 3.<\/p>\n<p id=\"fs-id1167836560949\">We do the same thing using function notation, the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa1fe37c29ef252a2729d235c875f0ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/> can be written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be88bdda715f166de8f126f328be8b94_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/> To find the value when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e211b3d19a6898a4c9192f117c1fe08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> we write:<\/p>\n<table id=\"fs-id1167836326537\" class=\"unnumbered unstyled\" summary=\"f of x = 4 x minus 5. Let x = 2. f of 2 = 4 times 2 minus 5. f of 2 = 3.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836447887\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d11400eca30b4e5ab418bd61ede75fa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836314769\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836399865\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836456485\">The value of the function at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> is 3.<\/p>\n<p id=\"fs-id1167836629682\">This process of finding the value of <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;\" \/> for a given value of <em data-effect=\"italics\">x<\/em> is called <em data-effect=\"italics\">evaluating the function.<\/em><\/p>\n<div data-type=\"example\" id=\"fs-id1167836521479\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836596680\">\n<div data-type=\"problem\" id=\"fs-id1167833224521\">\n<p id=\"fs-id1167833369207\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acb4328b3d8b3a3ec9913251ca5663e0_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;&#120;&#45;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/> evaluate the function.<\/p>\n<p id=\"fs-id1167836416554\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7454c1320ac593411a470ad12380405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" 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-dca5d7bc047aaa23a5ac85a2c257c5f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" 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-c25cd68506d88bf80f213795b26c27f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829756296\">\n<p id=\"fs-id1167829590752\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836507536\" class=\"unnumbered unstyled\" summary=\"f of x = 2 x squared plus 3 x minus 1. To evaluate f of 3, substitute 3 for x. f of 3 = 2 times 3 squared plus 3 times 3 minus 1. Simplify. f of 3 = 2 times 9 plus 3 times 3 minus 1. f of 3 = 18 plus 9 minus 1. f of 3 = 26.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836293439\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">To evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a41062e7d257e4d8b796525392940df8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/> substitute 3 for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832999716\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836520086\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832980523\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829852974\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836775152\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829740069\" class=\"unnumbered unstyled\" summary=\"f of x = 2 x squared plus 3 x minus 1. To evaluate f of negative 2, substitute negative 2 for x. f of negative 2 = 2 times the quantity negative 2 in parentheses squared plus 3 times negative 2 minus 1. Simplify. f of negative 2 = 2 times 4 plus negative 6 minus 1. f of negative 2 = 1.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/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_03_05_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792372816\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829694507\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829715540\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836532685\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829695370\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_016f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836601841\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836600305\" class=\"unnumbered unstyled\" summary=\"f of x = 2 x squared plus 3 x minus 1. To evaluate f of a, substitute a for x. f of a = 2 times a squared plus 3 times a minus 1. Simplify. f of a = 2 a squared plus 3 a minus 1.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836685384\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_017a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">To evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0ddd7617068c1b5df57c7142e93f06d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/> substitute <em data-effect=\"italics\">a<\/em> for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836362257\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836320903\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_017c_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-id1167836388368\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836648690\">\n<div data-type=\"problem\" id=\"fs-id1167836520515\">\n<p id=\"fs-id1167836363586\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7bc286652c1876d3f43b14ab2430e66_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;&#50;&#125;&#45;&#50;&#120;&#43;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/> evaluate the function.<\/p>\n<p id=\"fs-id1167836625930\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7454c1320ac593411a470ad12380405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" 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-134041031ac2f255a6139c40c1ff81ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" 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-f86f9045769dd7c509b16550f430662a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833135306\">\n<p id=\"fs-id1167825884792\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e1d84f2a4ce5504ff952a990caa7641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" 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-b3057788f9867c2a883ca27e50ceeeaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" 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-0c5d72789eea2256fd4a9636a6668433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#116;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824578714\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836729386\">\n<div data-type=\"problem\" id=\"fs-id1167833269886\">\n<p id=\"fs-id1167833269888\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-526656c3938108c84d363a8231bd9d2c_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;&#52;&#120;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/> evaluate the function.<\/p>\n<p id=\"fs-id1167829579645\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a73d5ba7eb27f0c2b0aa8c3e74588a43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" 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-323b7bb2b4781d94070ede4656b35c82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" 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-12317a46be480d10938fe8d98f2652fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829783753\">\n<p id=\"fs-id1167836732807\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1eac9f41e35dc3cf05c2b898fdd1f6aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" 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-fdc3c4c9e413fcb8bcc6942a5e1ec7fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" 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-17f0787bf89eb96186927978831ca799_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#104;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836518536\">In the last example, we found <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;\" \/> for a constant value of <em data-effect=\"italics\">x<\/em>. In the next example, we are asked to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d629b05700538f1d987aa1572837c00_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/> with values of <em data-effect=\"italics\">x<\/em> that are variables. We still follow the same procedure and substitute the variables in for the <em data-effect=\"italics\">x<\/em>.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829859398\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833137652\">\n<div data-type=\"problem\" id=\"fs-id1167836619820\">\n<p id=\"fs-id1167836619822\">For the function <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;\" \/> evaluate the function.<\/p>\n<p id=\"fs-id1167836320931\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06e516d23295f2f07fa0a27bcbe2b730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"44\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e04ee29807e4f441d57f163272a5fc44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" 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-4a3f29e9f8f03c5349de5ad304423c69_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;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833397154\">\n<p id=\"fs-id1167836754871\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1171790386499\" class=\"unnumbered unstyled\" summary=\"g of x = 3 x minus 5. To evaluate g of h squared, substitute h squared for x. g of h squared = 3 times h squared minus 5. Simplify. g of h squared = 3 h squared minus 5.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792580809\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_018a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">To evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcbdd5cad6b4e4d5739bde297f126541_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"53\" style=\"vertical-align: -7px;\" \/> substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae7704360a2632d9b1ffd59eb4541a89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#104;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171790163372\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_018b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792588637\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_018c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836556251\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1171792580965\" class=\"unnumbered unstyled\" summary=\"g of x = 3 x minus 5. To evaluate g of the quantity x plus 2, substitute x plus 2 for x. g of x plus 2 = 3 times the quantity x plus 2 in parentheses minus 5. Simplify. g of x plus 2 = 3 x plus 6 minus 5. g of x plus 2 = 3 x plus 1.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792545190\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_019a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">To evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c73ea2c9d57e62efce91b65b2f46886_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/> for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792543055\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_019b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171792802121\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_019c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171790304229\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_019d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836448921\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836606104\" class=\"unnumbered unstyled\" summary=\"g of x = 3 x minus 5. To evaluate g of x plus g of 2, first find g of 2. g of 2 = 3 times 2 minus 5. g of 2 = 1. Now find g of x plus g of 2. g of x plus g of 2 = 3 x minus 5 plus 1. The 3 x minus 5 is from g of x and the 1 is from the g of 2. Simplify. g of x plus g of 2 = 3 x minus 4.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829839522\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">To evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a97e216ca025cc84397ac80eb3f2901d_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;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/> first find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d648c116745c7ac45af1bac8e06f1bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836635207\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829685715\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836693279\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836322927\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td><\/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_03_05_020f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836539801\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_020g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167832981642\">Notice the difference between part <span class=\"token\">\u24d1<\/span> and <span class=\"token\">\u24d2<\/span>. We get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-343f43083b61165a630031cad04ae6ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ea01ee22792353d31cae561152d4c6a_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;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"170\" style=\"vertical-align: -4px;\" \/> So we see that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab1729872f108075fd59ca22c91e8665_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#110;&#101;&#32;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836730594\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832940643\">\n<div data-type=\"problem\" id=\"fs-id1167829788685\">\n<p id=\"fs-id1167829788687\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8d4c08a4b1f9107eb71463edcae05e5_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;&#55;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> evaluate the function.<\/p>\n<p id=\"fs-id1167829906594\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e38c652325725727d49e6dd1e5a2946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"49\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ada3bfcc9b358abbf0968ca07f63341e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" 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-4800833e0163cb9b08ea01345fca10c2_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;&#45;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836624292\">\n<p id=\"fs-id1167836554185\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cb8b70ab58be302b9a2eabc1f2c2abc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" 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-51138258f3aaed82151166349505f1e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"59\" 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-8d777eb5e2ca8bde3dc72ceda4556388_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167825766170\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829850927\">\n<p id=\"fs-id1167836293432\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fbefd90ce902617524fde4ec2aa53b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#43;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/> evaluate the function.<\/p>\n<p id=\"fs-id1167823012018\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8558dae9202bcdb4fb1bd74bc80dff8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"45\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4997600abe41f9d3f695b4b30a17197_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" 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-d39e555ee9adc3698371e3c381b22379_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829743779\">\n<p id=\"fs-id1167829717717\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dac10b7ae9d95be365ee3f5d9f076fea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"56\" 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-9d082aab86dbda3f149213325b8d3fe1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"50\" 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-08a71fdd8b90e0df7894e765e5aafae4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"50\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836283159\">Many everyday situations can be modeled using functions.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833158753\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833158755\">\n<div data-type=\"problem\" id=\"fs-id1167836533836\">\n<p id=\"fs-id1167836533838\">The number of unread emails in Sylvia\u2019s account is 75. This number grows by 10 unread emails a day. The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-463b3880d5c956c07c6e77a12969262c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#53;&#43;&#49;&#48;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/> represents the relation between the number of emails, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in days.<\/p>\n<p id=\"fs-id1167836442406\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p>\n<p id=\"fs-id1167833061546\"><span class=\"token\">\u24d1<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd0211c42a0d31ee9385be22888f7bb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/> Explain what this result means.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829715383\">\n<p id=\"fs-id1167829715385\"><span class=\"token\">\u24d0<\/span> The number of unread emails is a function of the number of days. The number of unread emails, <em data-effect=\"italics\">N<\/em>, depends on the number of days, <em data-effect=\"italics\">t<\/em>. Therefore, the variable <em data-effect=\"italics\">N<\/em>, is the dependent variable and the variable <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> is the independent variable.<\/p>\n<p id=\"fs-id1167833347223\"><span class=\"token\">\u24d1<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd0211c42a0d31ee9385be22888f7bb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/> Explain what this result means.<\/p>\n<table id=\"fs-id1167829709312\" class=\"unnumbered unstyled\" summary=\"N of t = 75 plus 10 t. Substitute in t = 5. N of 5 = 75 plus 10 times 5. Simplify. N of 5 = 75 plus 50. N of 5 = 125.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829850506\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Substitute in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fea44d192aff0ce938599188dbef067_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832971244\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_021b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833270224\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_021c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833309949\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_021d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167833413059\">Since 5 is the number of days, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2a06c15a3b3b1b17e0bf248d88a955e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/> is the number of unread emails after 5 days. After 5 days, there are 125 unread emails in the account.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836507876\">\n<p id=\"fs-id1167836507878\">The number of unread emails in Bryan\u2019s account is 100. This number grows by 15 unread emails a day. The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69c9b2b524e33d04ad5e03219c1ca4f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#48;&#48;&#43;&#49;&#53;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/> represents the relation between the number of emails, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in days.<\/p>\n<p id=\"fs-id1167836360895\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p>\n<p id=\"fs-id1167829619314\"><span class=\"token\">\u24d1<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c7eea926e66ea85b65127db573b3c35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/> Explain what this result means.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836622270\">\n<p id=\"fs-id1167836622272\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">t<\/em> IND; <em data-effect=\"italics\">N<\/em> DEP <span class=\"token\">\u24d1<\/span> 205; the number of unread emails in Bryan\u2019s account on the seventh day.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836481611\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829786754\">\n<div data-type=\"problem\" id=\"fs-id1167824735581\">\n<p id=\"fs-id1167824735583\">The number of unread emails in Anthony\u2019s account is 110. This number grows by 25 unread emails a day. The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd93ba7a36f83c18041033c94e49167e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#49;&#48;&#43;&#50;&#53;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/> represents the relation between the number of emails, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in days.<\/p>\n<p id=\"fs-id1167829743199\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p>\n<p id=\"fs-id1167836575515\"><span class=\"token\">\u24d1<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85571c99c0f41abf5e27ecf76d02f9ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -4px;\" \/> Explain what this result means.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836656679\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">t<\/em> IND; <em data-effect=\"italics\">N<\/em> DEP <span class=\"token\">\u24d1<\/span> 460; the number of unread emails in Anthony\u2019s account on the fourteenth day<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833128952\" class=\"media-2\">\n<p id=\"fs-id1167836508744\">Access this online resource for additional instruction and practice with relations and functions.<\/p>\n<ul id=\"fs-id1167836508748\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37introfunction\">Introduction to Functions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829711772\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836294674\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Function Notation:<\/strong> For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70ecad17068db077c0b28295e4626839_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/>\n<ul id=\"fs-id1167829810537\" data-bullet-style=\"open-circle\">\n<li><em data-effect=\"italics\">f<\/em> is the name of the function<\/li>\n<li><em data-effect=\"italics\">x<\/em> is the domain value<\/li>\n<li><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 the range value <em data-effect=\"italics\">y<\/em> corresponding to the value <em data-effect=\"italics\">x<\/em>\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-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;\" \/> as <em data-effect=\"italics\">f<\/em> of <em data-effect=\"italics\">x<\/em> or the value of <em data-effect=\"italics\">f<\/em> at <em data-effect=\"italics\">x<\/em>.<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Independent and Dependent Variables:<\/strong> For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-220367de6dc72b963d97846ff4643d95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -4px;\" \/>\n<ul id=\"fs-id1167836715045\" data-bullet-style=\"open-circle\">\n<li><em data-effect=\"italics\">x<\/em> is the independent variable as it can be any value in the domain<\/li>\n<li><em data-effect=\"italics\">y<\/em> is the dependent variable as its value depends on <em data-effect=\"italics\">x<\/em><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167826170977\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167826189010\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167833041789\"><strong data-effect=\"bold\">Find the Domain and Range of a Relation<\/strong><\/p>\n<p id=\"fs-id1167836701331\">In the following exercises, for each relation <span class=\"token\">\u24d0<\/span> find the domain of the relation <span class=\"token\">\u24d1<\/span> find the range of the relation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836694560\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833129254\">\n<p id=\"fs-id1167833129256\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-621e5a19d5d2ad63237b54cf89630ace_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;&#52;&#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;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"283\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836615948\">\n<p id=\"fs-id1167836615950\"><span class=\"token\">\u24d0<\/span> {1, 2, 3, 4, 5} <span class=\"token\">\u24d1<\/span> {4, 8, 12, 16, 20}<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829738657\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836729077\">\n<p id=\"fs-id1167836729080\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1822a47a410733d0ced3a589d0294b28_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;&#45;&#50;&#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;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"334\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836289174\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836289176\">\n<p id=\"fs-id1167833021591\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b472b1a25453c9318e365323579dcab_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;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#57;&#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;&#50;&#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=\"298\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829713369\">\n<p id=\"fs-id1167829713371\"><span class=\"token\">\u24d0<\/span> {1, 5, 7, \u22122} <span class=\"token\">\u24d1<\/span> {7, 3, 9, \u22123, 8}<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836513003\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836686245\">\n<p id=\"fs-id1167836686247\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd9be7443c8e3af4a20ca04fe017b9dd_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;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#49;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#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=\"329\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836309438\">In the following exercises, use the mapping of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation, <span class=\"token\">\u24d1<\/span> find the domain of the relation, and <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167824736897\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829693414\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829693415\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cRebecca\u201d, \u201cJennifer\u201d, \u201cJohn\u201d, \u201cHector\u201d, \u201cLuis\u201d, \u201cEbony\u201d, \u201cRaphael\u201d, \u201cMeredith\u201d, \u201cKaren\u201d, and \u201cJoseph\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 18\u201d, \u201cFebruary 15\u201d, \u201cApril 1\u201d, \u201cApril 7\u201d, \u201cJune 23\u201d, \u201cJuly 30\u201d, \u201cAugust 19\u201d, and \u201cNovember 6\u201d. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Rebecca to January 18. The second arrow goes from Jennifer to April 1. The third arrow goes from John to January 18. The fourth arrow goes from Hector to June 23. The fifth arrow goes from Luis to February 15. The sixth arrow goes from Ebony to April 7. The seventh arrow goes from Raphael to November 6. The eighth arrow goes from Meredith to August 19. The ninth arrow goes from Karen to August 19. The tenth arrow goes from Joseph to July 30.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cRebecca\u201d, \u201cJennifer\u201d, \u201cJohn\u201d, \u201cHector\u201d, \u201cLuis\u201d, \u201cEbony\u201d, \u201cRaphael\u201d, \u201cMeredith\u201d, \u201cKaren\u201d, and \u201cJoseph\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 18\u201d, \u201cFebruary 15\u201d, \u201cApril 1\u201d, \u201cApril 7\u201d, \u201cJune 23\u201d, \u201cJuly 30\u201d, \u201cAugust 19\u201d, and \u201cNovember 6\u201d. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Rebecca to January 18. The second arrow goes from Jennifer to April 1. The third arrow goes from John to January 18. The fourth arrow goes from Hector to June 23. The fifth arrow goes from Luis to February 15. The sixth arrow goes from Ebony to April 7. The seventh arrow goes from Raphael to November 6. The eighth arrow goes from Meredith to August 19. The ninth arrow goes from Karen to August 19. The tenth arrow goes from Joseph to July 30.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836558109\">\n<p id=\"fs-id1167833369090\"><span class=\"token\">\u24d0<\/span> (Rebecca, January 18), (Jennifer, April 1), (John, January 18), (Hector, June 23), (Luis, February 15), (Ebony, April 7), (Raphael, November 6), (Meredith, August 19), (Karen, August 19), (Joseph, July 30)<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> {Rebecca, Jennifer, John, Hector, Luis, Ebony, Raphael, Meredith, Karen, Joseph}<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> {January 18, April 1, June 23, February 15, April 7, November 6, August 19, July 30}<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836600990\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829614614\">\n<p id=\"fs-id1167829614616\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829614618\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cAmy\u201d, \u201cCarol\u201d, \u201cDevon\u201d, \u201cHarrison\u201d, \u201cJackson\u201d, \u201cLabron\u201d, \u201cMason\u201d, \u201cNatalie\u201d, \u201cPaul\u201d, and \u201cSylvester\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 5\u201d, \u201cJanuary 7\u201d, \u201cFebruary 14\u201d, \u201cMarch 1\u201d, \u201cApril 7\u201d, \u201cMay 30\u201d, \u201cJuly 20\u201d, \u201cAugust 1\u201d, \u201cNovember 13\u201d, and \u201cNovember 26\u201d. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Amy to February 14. The second arrow goes from Carol to May 30. The third arrow goes from Devon to January 5. The fourth arrow goes from Harrison to January 7. The fifth arrow goes from Jackson to November 26. The sixth arrow goes from Labron to April 7. The seventh arrow goes from Mason to July 20. The eighth arrow goes from Natalie to March 1. The ninth arrow goes from Paul to August 1. The tenth arrow goes from Sylvester to November 13.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cAmy\u201d, \u201cCarol\u201d, \u201cDevon\u201d, \u201cHarrison\u201d, \u201cJackson\u201d, \u201cLabron\u201d, \u201cMason\u201d, \u201cNatalie\u201d, \u201cPaul\u201d, and \u201cSylvester\u201d. The table on the right has the header \u201cBirthday\u201d and lists the dates \u201cJanuary 5\u201d, \u201cJanuary 7\u201d, \u201cFebruary 14\u201d, \u201cMarch 1\u201d, \u201cApril 7\u201d, \u201cMay 30\u201d, \u201cJuly 20\u201d, \u201cAugust 1\u201d, \u201cNovember 13\u201d, and \u201cNovember 26\u201d. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Amy to February 14. The second arrow goes from Carol to May 30. The third arrow goes from Devon to January 5. The fourth arrow goes from Harrison to January 7. The fifth arrow goes from Jackson to November 26. The sixth arrow goes from Labron to April 7. The seventh arrow goes from Mason to July 20. The eighth arrow goes from Natalie to March 1. The ninth arrow goes from Paul to August 1. The tenth arrow goes from Sylvester to November 13.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833128978\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833128980\">\n<p id=\"fs-id1167836511320\">For a woman of height <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38b5e59e580ffbefbca99a00a24b9a33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#52;&#8243;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -1px;\" \/> the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-230933c608426efccb8170c44f61ee0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#46;&#53;&#45;&#50;&#52;&#46;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -1px;\" \/> is considered healthy.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829850494\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cWeight (lbs)\u201d and lists the numbers plus 100, 110, 120, 130, 140, 150, and 160. The table on the right has the header \u201cBMI\u201d and lists the numbers 18. 9, 22. 3, 17. 2, 24. 0, 25. 7, 20. 6, and 27. 5. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from plus 100 to 17. 2. The second arrow goes from 110 to 18. 9. The third arrow goes from 120 to 20. 6. The fourth arrow goes from 130 to 22. 3. The fifth arrow goes from 140 to 24. 0. The sixth arrow goes from 150 to 25. 7. The seventh arrow goes from 160 to 27. 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cWeight (lbs)\u201d and lists the numbers plus 100, 110, 120, 130, 140, 150, and 160. The table on the right has the header \u201cBMI\u201d and lists the numbers 18. 9, 22. 3, 17. 2, 24. 0, 25. 7, 20. 6, and 27. 5. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from plus 100 to 17. 2. The second arrow goes from 110 to 18. 9. The third arrow goes from 120 to 20. 6. The fourth arrow goes from 130 to 22. 3. The fifth arrow goes from 140 to 24. 0. The sixth arrow goes from 150 to 25. 7. The seventh arrow goes from 160 to 27. 5.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836410493\">\n<p id=\"fs-id1167836410496\"><span class=\"token\">\u24d0<\/span> (+100, 17. 2), (110, 18.9), (120, 20.6), (130, 22.3), (140, 24.0), (150, 25.7), (160, 27.5) <span class=\"token\">\u24d1<\/span> {+100, 110, 120, 130, 140, 150, 160,} <span class=\"token\">\u24d2<\/span> {17.2, 18.9, 20.6, 22.3, 24.0, 25.7, 27.5}<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829590523\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829590525\">\n<p id=\"fs-id1167829590527\">For a man of height <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1483bc5d0c7ebda6efe5f87d2d5290ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#49;&#49;&#92;&#112;&#114;&#105;&#109;&#101;&#32;&#92;&#112;&#114;&#105;&#109;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"42\" style=\"vertical-align: -1px;\" \/> the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-230933c608426efccb8170c44f61ee0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#46;&#53;&#45;&#50;&#52;&#46;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -1px;\" \/> is considered healthy.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167836328549\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cWeight (lbs)\u201d and lists the numbers 130, 140, 150, 160, 170, 180, 190, and 200. The table on the right has the header \u201cBMI\u201d and lists the numbers 22. 3, 19. 5, 20. 9, 27. 9, 25. 1, 26. 5, 23. 7, and 18. 1. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from 130 to 18. 1. The second arrow goes from 140 to 19. 5. The third arrow goes from 150 to 20. 9. The fourth arrow goes from 160 to 22. 3. The fifth arrow goes from 170 to 23. 7. The sixth arrow goes from 180 to 25. 1. The seventh arrow goes from 190 to 26. 5. The eighth arrow goes from 200 to 27. 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cWeight (lbs)\u201d and lists the numbers 130, 140, 150, 160, 170, 180, 190, and 200. The table on the right has the header \u201cBMI\u201d and lists the numbers 22. 3, 19. 5, 20. 9, 27. 9, 25. 1, 26. 5, 23. 7, and 18. 1. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from 130 to 18. 1. The second arrow goes from 140 to 19. 5. The third arrow goes from 150 to 20. 9. The fourth arrow goes from 160 to 22. 3. The fifth arrow goes from 170 to 23. 7. The sixth arrow goes from 180 to 25. 1. The seventh arrow goes from 190 to 26. 5. The eighth arrow goes from 200 to 27. 9.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167825791209\">In the following exercises, use the graph of the relation to <span class=\"token\">\u24d0<\/span> list the ordered pairs of the relation <span class=\"token\">\u24d1<\/span> find the domain of the relation <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836621459\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836623116\">\n<p id=\"fs-id1167836623118\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836623119\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 1), (0, negative 3), (2, 3), (4, negative 1), and (4, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 1), (0, negative 3), (2, 3), (4, negative 1), and (4, negative 3).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836448402\">\n<p id=\"fs-id1167836448404\"><span class=\"token\">\u24d0<\/span> (2, 3), (4, \u22123), (\u22122, \u22121), (\u22123, 4), (4, \u22121), (0, \u22123) <span class=\"token\">\u24d1<\/span> {\u22123, \u22122, 0, 2, 4}<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> {\u22123, \u22121, 3, 4}<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833274699\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833274701\">\n<p id=\"fs-id1167836599639\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836599640\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 4), (negative 2, 0), (negative 1, 3), (1, 5), and (4, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 4), (negative 3, negative 4), (negative 2, 0), (negative 1, 3), (1, 5), and (4, negative 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836707143\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836524480\">\n<p id=\"fs-id1167836524482\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836524483\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 1, 4), (negative 1, negative 4), (0, 3), (0, negative 3), (1, 4), and (1, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_207_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 1, 4), (negative 1, negative 4), (0, 3), (0, negative 3), (1, 4), and (1, negative 4).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829861860\">\n<p id=\"fs-id1167829861862\"><span class=\"token\">\u24d0<\/span> (1, 4), (1, \u22124), (\u22121, 4), (\u22121, \u22124), (0, 3), (0, \u22123) <span class=\"token\">\u24d1<\/span> {\u22121, 0, 1} <span class=\"token\">\u24d2<\/span> {\u22124, \u22123, 3,4}<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836509162\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836509164\">\n<p id=\"fs-id1167836546295\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836546296\" data-alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The points (negative 2, negative 6), (negative 2, negative 3), (0, 0), (0. 5, 1. 5), (1, 3), and (3, 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_208_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The points (negative 2, negative 6), (negative 2, negative 3), (0, 0), (0. 5, 1. 5), (1, 3), and (3, 6).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167836513559\"><strong data-effect=\"bold\">Determine if a Relation is a Function<\/strong><\/p>\n<p id=\"fs-id1167833060894\">In the following exercises, use the set of ordered pairs to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function, <span class=\"token\">\u24d1<\/span> find the domain of the relation, and <span class=\"token\">\u24d2<\/span> find the range of the relation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829666517\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833059308\">\n<p id=\"fs-id1167833059310\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3402e18e5f0e110f846d4759aa31d52_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"196\" 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-910613f291e33fa415d78626ac5ab9bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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=\"18\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833137411\">\n<p id=\"fs-id1167833137413\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> {\u22123, \u22122, \u22121, 0, 1, 2, 3} <span class=\"token\">\u24d2<\/span> {9, 4, 1, 0}<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836688996\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836688998\">\n<p id=\"fs-id1167836689000\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5118effbf3594019197b423dd20123c7_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"196\" 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-29a839f0e2413e990a530b8495561c23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829745706\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829745708\">\n<p id=\"fs-id1167829745710\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b92f8ea5a47b401913bb00a38b481e3b_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"205\" 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-3d6423c591135d1b7df4a8ac414aee2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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=\"18\" width=\"197\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836529454\">\n<p id=\"fs-id1167829853364\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> {\u22123, \u22122, \u22121, 0, 1, 2, 3} <span class=\"token\">\u24d2<\/span> 0, 1, 8, 27}<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833138708\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833138710\">\n<p id=\"fs-id1167826077064\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34efc4900176406f73fb1eceaf060ff4_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;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#56;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"246\" 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-3d6423c591135d1b7df4a8ac414aee2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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=\"18\" width=\"197\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167836552752\">In the following exercises, use the mapping to <span class=\"token\">\u24d0<\/span> determine whether the relation is a function, <span class=\"token\">\u24d1<\/span> find the domain of the function, and <span class=\"token\">\u24d2<\/span> find the range of the function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829742787\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829742789\">\n<p id=\"fs-id1167829742791\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167833056754\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNumber\u201d and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header \u201cAbsolute Value\u201d and lists the numbers 0, 1, 2, and 3. There are arrows starting at numbers in the number table and pointing towards numbers in the absolute value table. The first arrow goes from negative 3 to 3. The second arrow goes from negative 2 to 2. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 2. The seventh arrow goes from 3 to 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_209_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNumber\u201d and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header \u201cAbsolute Value\u201d and lists the numbers 0, 1, 2, and 3. There are arrows starting at numbers in the number table and pointing towards numbers in the absolute value table. The first arrow goes from negative 3 to 3. The second arrow goes from negative 2 to 2. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 2. The seventh arrow goes from 3 to 3.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829749063\">\n<p id=\"fs-id1167829749065\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> {\u22123, \u22122, \u22121, 0, 1, 2, 3} <span class=\"token\">\u24d2<\/span> {0, 1, 2, 3}<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829741614\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829741616\">\n<p id=\"fs-id1167829908273\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829908274\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNumber\u201d and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header \u201cSquare\u201d and lists the numbers 0, 1, 4, and 9. There are arrows starting at numbers in the number table and pointing towards numbers in the square table. The first arrow goes from negative 3 to 9. The second arrow goes from negative 2 to 4. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 4. The seventh arrow goes from 3 to 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_210_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cNumber\u201d and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header \u201cSquare\u201d and lists the numbers 0, 1, 4, and 9. There are arrows starting at numbers in the number table and pointing towards numbers in the square table. The first arrow goes from negative 3 to 9. The second arrow goes from negative 2 to 4. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 4. The seventh arrow goes from 3 to 9.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836697708\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836697710\">\n<p id=\"fs-id1167832930330\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cJenny\u201d, \u201cR and y\u201d, \u201cDennis\u201d, \u201cEmily\u201d, and \u201cRaul\u201d. The table on the right has the header \u201cEmail\u201d and lists the email addresses RHern and ez@state. edu, JKim@gmail.com, Raul@gmail.com, ESmith@state. edu, DBrown@aol.com, jenny@aol.com, and R and y@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jenny to JKim@gmail.com. The second arrow goes from Jenny to jenny@aol.com. The third arrow goes from R and y to R and y@gmail.com. The fourth arrow goes from Dennis to DBrown@aol.com. The fifth arrow goes from Emily to ESmith@state. edu. The sixth arrow goes from Raul to RHern and ez@state. edu. The seventh arrow goes from Raul to Raul@gmail.com.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_211_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cJenny\u201d, \u201cR and y\u201d, \u201cDennis\u201d, \u201cEmily\u201d, and \u201cRaul\u201d. The table on the right has the header \u201cEmail\u201d and lists the email addresses RHern and ez@state. edu, JKim@gmail.com, Raul@gmail.com, ESmith@state. edu, DBrown@aol.com, jenny@aol.com, and R and y@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jenny to JKim@gmail.com. The second arrow goes from Jenny to jenny@aol.com. The third arrow goes from R and y to R and y@gmail.com. The fourth arrow goes from Dennis to DBrown@aol.com. The fifth arrow goes from Emily to ESmith@state. edu. The sixth arrow goes from Raul to RHern and ez@state. edu. The seventh arrow goes from Raul to Raul@gmail.com.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167824737653\">\n<p><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> {Jenny, R and y, Dennis, Emily, Raul} <span class=\"token\">\u24d2<\/span> {RHern and ez@state.edu, JKim@gmail.com, Raul@gmail.com, ESmith@state.edu, DBroen@aol.com, jenny@aol.cvom, R and y@gmail.com}<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833024527\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833256809\">\n<p id=\"fs-id1167833256811\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167833256812\" data-alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cJon\u201d, \u201cRachel\u201d, \u201cMatt\u201d, \u201cLeslie\u201d, \u201cChris\u201d, \u201cBeth\u201d, and \u201cLiz\u201d. The table on the right has the header \u201cEmail\u201d and lists the email addresses chrisg@gmail.com, lizzie@aol.com, jong@gmail.com, mattg@gmail.com, Rachel@state. edu, leslie@aol.com, and bethc@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jon to jong@gmail.com. The second arrow goes from Rachel to Rachel@state. edu. The third arrow goes from Matt to mattg@gmail.com. The fourth arrow goes from Leslie to leslie@aol.com. The fifth arrow goes from Chris to chrisg@gmail.com. The sixth arrow goes from Beth to bethc@gmail.com. The seventh arrow goes from Liz to lizzie@aol.com.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_212_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two table that each have one column. The table on the left has the header \u201cName\u201d and lists the names \u201cJon\u201d, \u201cRachel\u201d, \u201cMatt\u201d, \u201cLeslie\u201d, \u201cChris\u201d, \u201cBeth\u201d, and \u201cLiz\u201d. The table on the right has the header \u201cEmail\u201d and lists the email addresses chrisg@gmail.com, lizzie@aol.com, jong@gmail.com, mattg@gmail.com, Rachel@state. edu, leslie@aol.com, and bethc@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jon to jong@gmail.com. The second arrow goes from Rachel to Rachel@state. edu. The third arrow goes from Matt to mattg@gmail.com. The fourth arrow goes from Leslie to leslie@aol.com. The fifth arrow goes from Chris to chrisg@gmail.com. The sixth arrow goes from Beth to bethc@gmail.com. The seventh arrow goes from Liz to lizzie@aol.com.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167836705602\">In the following exercises, determine whether each equation is a function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836705606\">\n<div data-type=\"problem\" id=\"fs-id1167836525998\">\n<p id=\"fs-id1167836526000\">\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-d7bb26d3c9954af84f4dec11d7baa9fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#121;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" 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-f0e2ebf2a1b63bd75c0c19696a700e4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" 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-df20cded9c40b0e5a06ef05354cbdf88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836609321\">\n<p id=\"fs-id1167836650178\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> yes <span class=\"token\">\u24d2<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836480280\">\n<div data-type=\"problem\" id=\"fs-id1167829828558\">\n<p id=\"fs-id1167829828560\">\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-47bdbf719e41e4cd61a7a609f53c9186_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" 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-4b1d53b124475c894e7d7504e830fe49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" 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-7ea82467800eb09d48a96941a961ddb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836526018\">\n<div data-type=\"problem\" id=\"fs-id1167836526020\">\n<p id=\"fs-id1167836526022\">\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-1f4cbb0bf05a1aa6911908bbed3c4dd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" 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-f546a2df09d32fb7ec25f52dfe9f13fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" 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-4fa0b37b12e2ed9b6d126e5523cd6dbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#50;&#121;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833135088\">\n<p id=\"fs-id1167833135090\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no <span class=\"token\">\u24d2<\/span> yes<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833212947\">\n<div data-type=\"problem\" id=\"fs-id1167833212949\">\n<p id=\"fs-id1167836521932\">\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-b17d88f3ef2c272cc2a5f53f5d8888a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#52;&#121;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" 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-4d8ff4387242026ed080f28c74170259_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" 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-4d0548eaff90dbc6255d843c6d16075c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167836714017\"><strong data-effect=\"bold\">Find the Value of a Function<\/strong><\/p>\n<p id=\"fs-id1167829717540\">In the following exercises, evaluate the function: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a73d5ba7eb27f0c2b0aa8c3e74588a43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" 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-134041031ac2f255a6139c40c1ff81ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" 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-4153ddea4dfdd594c6187304119af999_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167823013283\">\n<div data-type=\"problem\" id=\"fs-id1167823013285\">\n<p id=\"fs-id1167823013287\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7d7f35c2399174a9ddcfd9a5a2dab44_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829624155\">\n<p id=\"fs-id1167829624157\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dce4188fc533644dec2bee74fb0cdfcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" 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-392cb53ec449c4fbc0c0a2774aba305c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" 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-0d19714388c211db9126754d919a9b84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#97;&#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\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829747304\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4496bc44aea2b734c9eca2a432647155_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;&#43;&#52;\" 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-id1167829744293\">\n<div data-type=\"problem\" id=\"fs-id1167829744295\">\n<p id=\"fs-id1167836423791\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e2a850d8e2941c172d598caac74bd92_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;&#52;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833047067\">\n<p id=\"fs-id1167836690281\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26a200f690989415b090eda2dfd7953a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#54;\" 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-b3057788f9867c2a883ca27e50ceeeaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" 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-ca06e526533ed43ddc134c7f3a7936d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#97;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836391029\">\n<div data-type=\"problem\" id=\"fs-id1167836391032\">\n<p id=\"fs-id1167836391034\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bf3c8a0935c81ecf85ebb11b5498b57_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;&#54;&#120;&#45;&#51;\" 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-id1167829877909\">\n<div data-type=\"problem\" id=\"fs-id1167836575834\">\n<p id=\"fs-id1167836575836\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04ccbbff3314cb043b0c57622d1ecc17_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;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824737178\">\n<p id=\"fs-id1167836692744\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-261985f21b464c0b0545377936a63280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" 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-80926bcf7fa3123a18162c5aeaa2a627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" 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-30d223ada160569fbf075fe1abd36d23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#97;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836787728\">\n<div data-type=\"problem\" id=\"fs-id1167829789882\">\n<p id=\"fs-id1167829789884\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa2ddd89b30ed0f562c0894883414d11_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;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836574347\">\n<div data-type=\"problem\" id=\"fs-id1167836574349\">\n<p id=\"fs-id1167829695807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50c27609703936c1899d95675fe14de2_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;&#45;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829689462\">\n<p id=\"fs-id1167829689464\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b63da481ad9e9d4564fbdda7b062dfae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" 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-b3057788f9867c2a883ca27e50ceeeaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" 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-226e2b3ee58607082c0128f8a986cd89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#97;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829877536\">\n<div data-type=\"problem\" id=\"fs-id1167836791925\">\n<p id=\"fs-id1167836791928\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-843f5d24062c993994ddb02e80c4c39f_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;&#50;&#125;&#43;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836575175\">In the following exercises, evaluate the function: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06e516d23295f2f07fa0a27bcbe2b730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"44\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e04ee29807e4f441d57f163272a5fc44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" 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-e2a710a17795a763d4c10ff143663bc7_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;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829595951\">\n<div data-type=\"problem\" id=\"fs-id1167829595953\">\n<p id=\"fs-id1167826132424\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aabdcbbebc2342e4cdf9513612595723_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;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836575320\">\n<p id=\"fs-id1167836575323\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d04a4da1f82463de96cf28c50728151_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -7px;\" \/><\/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-abeb37cf7518bba766a6696732f30817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" 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-140119ce7cdbfd8d8752e12471ccd3f2_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;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836399733\">\n<div data-type=\"problem\" id=\"fs-id1167836399735\">\n<p id=\"fs-id1167833208016\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1150f03a9c38af8a8cb014e3d773fdb_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;&#56;\" 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-id1167833054462\">\n<div data-type=\"problem\" id=\"fs-id1167833054464\">\n<p id=\"fs-id1167829651167\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3726f2edf46698e6a0b16672d946fbd_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;&#45;&#51;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829754538\">\n<p id=\"fs-id1167829754540\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50eba39b88f1fca5132e60d12c0d7517_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"140\" style=\"vertical-align: -7px;\" \/><\/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-46b22510ac2a114f4bd987af16306827_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#120;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" 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-8f1bac35f540a9716d6c56666e5efffa_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;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#120;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829924597\">\n<div data-type=\"problem\" id=\"fs-id1167829924599\">\n<p id=\"fs-id1167829924601\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1b8ae9452492829bc18a3a0b16d1071_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;&#45;&#56;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836613442\">\n<div data-type=\"problem\" id=\"fs-id1167836613444\">\n<p id=\"fs-id1167836613446\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2851308c6c35206680421488c51a494d_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;&#45;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824976320\">\n<p id=\"fs-id1167836553636\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d0c8dab0b6640b5d168879a0593c03e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#45;&#123;&#104;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"117\" style=\"vertical-align: -7px;\" \/><\/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-92c90242e79f8102f3ecd7bc339c6d4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#45;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" 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-7aaf580b1452e94bdfa2e4779214e8ba_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;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#45;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833381469\">\n<div data-type=\"problem\" id=\"fs-id1167829860145\">\n<p id=\"fs-id1167829860148\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7010758994141f04825695b599f4263c_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;&#55;&#45;&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826171384\">In the following exercises, evaluate the function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826025131\">\n<div data-type=\"problem\" id=\"fs-id1167826025133\">\n<p id=\"fs-id1167826025135\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97adcb94e7e1ed013f98b0d0edd69a72_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;&#50;&#125;&#45;&#53;&#120;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a73d5ba7eb27f0c2b0aa8c3e74588a43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829879609\">\n<p id=\"fs-id1167829879611\">2<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833240212\">\n<div data-type=\"problem\" id=\"fs-id1167833240214\">\n<p id=\"fs-id1167826206218\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9eb7cc5726f377353c4d2373d6fedc5_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;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2adbcd4624c8f3e37ee45ba3f66530b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829713382\">\n<div data-type=\"problem\" id=\"fs-id1167829713384\">\n<p id=\"fs-id1167829713386\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8c2a56201027ed936b767d0f00acf7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#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;&#43;&#49;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/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-ab702bb67c5a96b46350e837c6fdd8cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829651457\">\n<p id=\"fs-id1167829651459\">6<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833382456\">\n<div data-type=\"problem\" id=\"fs-id1167833382458\">\n<p id=\"fs-id1167833382460\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b251d38a5d10e55bac8228c321d72a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#50;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/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-565d1a252293960065fdda7713d1882d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833052040\">\n<div data-type=\"problem\" id=\"fs-id1167833052042\">\n<p id=\"fs-id1167833052044\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbd65ba4afa28a144bb632bae2538b3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#124;&#116;&#45;&#53;&#124;&#43;&#52;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93495ad7b580fb77c102c40fbc21a31f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824976354\">\n<p id=\"fs-id1167824976356\">22<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836649081\">\n<div data-type=\"problem\" id=\"fs-id1167836649083\">\n<p id=\"fs-id1167836649085\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c42c387b3caba2ef7773c80d2c944c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#124;&#121;&#45;&#49;&#124;&#45;&#51;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"153\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b64e0828407f65f5008a90d110bd683f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829906037\">\n<div data-type=\"problem\" id=\"fs-id1167829906040\">\n<p id=\"fs-id1167829906042\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c85caccd383b393066ef0cf84daa5092_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;&#43;&#50;&#125;&#123;&#120;&#45;&#49;&#125;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a73d5ba7eb27f0c2b0aa8c3e74588a43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829579006\">\n<p id=\"fs-id1167829579008\">4<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833309035\">\n<div data-type=\"problem\" id=\"fs-id1167833309037\">\n<p id=\"fs-id1167833309039\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e82d8214659da4d3c305e74253f9fd4f_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;&#50;&#125;&#123;&#120;&#43;&#50;&#125;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"93\" style=\"vertical-align: -8px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49fcaf9fe488cacd08d464d30c0c480c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836409828\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836409832\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833386164\">\n<p id=\"fs-id1167833386167\">The number of unwatched shows in Sylvia\u2019s DVR is 85. This number grows by 20 unwatched shows per week. The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49fb626aea9020963e108df9d779ccbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#53;&#43;&#50;&#48;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/> represents the relation between the number of unwatched shows, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in weeks.<\/p>\n<p id=\"fs-id1167824737629\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p>\n<p id=\"fs-id1167836697913\"><span class=\"token\">\u24d1<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3c7e18769f7f5a6ca3fd3ad9138d1d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/> Explain what this result means<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833378063\">\n<p id=\"fs-id1167832935494\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">t<\/em> IND; <em data-effect=\"italics\">N<\/em> DEP<\/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-37b6891fe9d16029d529039d848e740c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#54;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -4px;\" \/> the number of unwatched shows in Sylvia\u2019s DVR at the fourth week. <\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825884747\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167825884749\">\n<p id=\"fs-id1167825884751\">Every day a new puzzle is downloaded into Ken\u2019s account. Right now he has 43 puzzles in his account. The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7383da60189278f77d2fdc2f16493ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#51;&#43;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/> represents the relation between the number of puzzles, <em data-effect=\"italics\">N<\/em>, and the time, <em data-effect=\"italics\">t<\/em>, measured in days.<\/p>\n<p id=\"fs-id1167829620296\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p>\n<p id=\"fs-id1167829620303\"><span class=\"token\">\u24d1<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b3cc08efa09ba0444e8d94b4d419178_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -4px;\" \/> Explain what this result means.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833380107\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836574149\">\n<p id=\"fs-id1167836574151\">The daily cost to the printing company to print a book is modeled by the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0d1e0924dca9d74e3b3990822ca529f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#46;&#50;&#53;&#120;&#43;&#49;&#53;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"164\" style=\"vertical-align: -4px;\" \/> where <em data-effect=\"italics\">C<\/em> is the total daily cost and <em data-effect=\"italics\">x<\/em> is the number of books printed.<\/p>\n<p id=\"fs-id1167833196806\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p>\n<p id=\"fs-id1167826077595\"><span class=\"token\">\u24d1<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa781ca25072efd01360c691261b5f54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/> Explain what this result means.<\/p>\n<p id=\"fs-id1167829831994\"><span class=\"token\">\u24d2<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a55b09195b94b6127ba95d7c61b43a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/> Explain what this result means.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836406944\">\n<p id=\"fs-id1167836406946\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">x<\/em> IND; <em data-effect=\"italics\">C<\/em> DEP<\/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-dc480146c83c4e6c54f57cf8e92638a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#53;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> the daily cost if no books are printed<\/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-ca01c9cafdad444af00cbb04356ae8ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#55;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/> the daily cost of printing 1000 books<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829749356\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829749358\">\n<p id=\"fs-id1167829749361\">The daily cost to the manufacturing company is modeled by the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e547799a8a9f291d10b030996b430187_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#46;&#50;&#53;&#120;&#43;&#50;&#53;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"164\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-574e0c4bf0d6708c2c77394ebfc62f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/> is the total daily cost and <em data-effect=\"italics\">x<\/em> is the number of items manufactured.<\/p>\n<p id=\"fs-id1167829879667\"><span class=\"token\">\u24d0<\/span> Determine the independent and dependent variable.<\/p>\n<p id=\"fs-id1167833272975\"><span class=\"token\">\u24d1<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4900f9f13fe08e9fb11a302275abcded_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/> Explain what this result means.<\/p>\n<p id=\"fs-id1167836494341\"><span class=\"token\">\u24d2<\/span> Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-836062f99d9d04e185a2eb0a85220678_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> Explain what this result means.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167829756260\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167829756267\">\n<div data-type=\"problem\" id=\"fs-id1167829878963\">\n<p id=\"fs-id1167829878966\">In your own words, explain the difference between a relation and a function.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829878971\">\n<div data-type=\"problem\" id=\"fs-id1167833054106\">\n<p id=\"fs-id1167833054108\">In your own words, explain what is meant by domain and range.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833054114\">\n<div data-type=\"problem\" id=\"fs-id1167836755007\">\n<p id=\"fs-id1167836755009\">Is every relation a function? Is every function a relation?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836755014\">\n<div data-type=\"problem\" id=\"fs-id1167829718391\">\n<p id=\"fs-id1167829718394\">How do you find the value of a function?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829783756\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167829783762\"><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-id1167836607305\" data-alt=\"The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cconfidently\u201d, the third is \u201cwith some help\u201d, \u201cno minus I don\u2019t get it!\u201d. Under the first column are the phrases \u201cfind the domain and range of a relation\u201d, \u201cdetermine if a relation is a function\u201d, and \u201cfind the value of a function\u201d. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_05_213_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cconfidently\u201d, the third is \u201cwith some help\u201d, \u201cno minus I don\u2019t get it!\u201d. Under the first column are the phrases \u201cfind the domain and range of a relation\u201d, \u201cdetermine if a relation is a function\u201d, and \u201cfind the value of a function\u201d. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.\" \/><\/span><\/p>\n<p id=\"fs-id1167836607297\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167829808856\">\n<dt>domain of a relation<\/dt>\n<dd id=\"fs-id1167836729545\">The domain of a relation is all the <em data-effect=\"italics\">x<\/em>-values in the ordered pairs of the relation.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833327007\">\n<dt>function<\/dt>\n<dd id=\"fs-id1167833327010\">A function is a relation that assigns to each element in its domain exactly one element in the range.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833327015\">\n<dt>mapping<\/dt>\n<dd id=\"fs-id1167833382797\">A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833382803\">\n<dt>range of a relation<\/dt>\n<dd id=\"fs-id1167832978259\">The range of a relation is all the <em data-effect=\"italics\">y-<\/em>values in the ordered pairs of the relation.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833175472\">\n<dt>relation<\/dt>\n<dd id=\"fs-id1167833175475\">A relation is any set of ordered pairs,<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ab61ec2033ed5b69d0447eac5d6a4f8_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> All the <em data-effect=\"italics\">x<\/em>-values in the ordered pairs together make up the domain. All the <em data-effect=\"italics\">y<\/em>-values in the ordered pairs together make up the range.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2017","chapter","type-chapter","status-publish","hentry"],"part":1643,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2017","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\/2017\/revisions"}],"predecessor-version":[{"id":15154,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2017\/revisions\/15154"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/1643"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2017\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=2017"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=2017"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=2017"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=2017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}