{"id":773,"date":"2019-04-29T17:24:03","date_gmt":"2019-04-29T21:24:03","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=chapter&p=773"},"modified":"2019-12-05T15:54:22","modified_gmt":"2019-12-05T20:54:22","slug":"11-2-operations-on-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/chapter\/11-2-operations-on-functions\/","title":{"raw":"11.2 Operations on Functions","rendered":"11.2 Operations on Functions"},"content":{"raw":"[latexpage]\r\n\r\nIn Chapter 5, you solved systems of linear equations through substitution, addition, subtraction, multiplication, and division. A similar process is employed in this topic, where you will add, subtract, multiply, divide, or substitute functions. The notation used for this looks like the following:\r\n\r\nGiven two functions \\(f(x)\\) and \\(g(x)\\):\r\n

\\(\\begin{array}{clcl}\r\nf(x) + g(x)&\\text{ is the same as }&(f + g)(x)&\\text{ and means the addition of these two functions} \\\\\r\nf(x) - g(x)&\\text{ is the same as }&(f - g)(x)&\\text{ and means the subtraction of these two functions} \\\\\r\nf(x)\\cdot g(x)&\\text{ is the same as }&(f\\cdot g)(x)&\\text{ and means the multiplication of these two functions} \\\\\r\nf(x)\\div g(x)&\\text{ is the same as }&(f\\div g)(x)&\\text{ and means the addition of these two functions}\r\n\\end{array}\\)<\/p>\r\nWhen encountering questions about operations on functions, you will generally be asked to do two things: combine the equations in some described fashion and to substitute some value to replace the variable in the original equation. These are illustrated in the following examples.\r\n

\r\n

Example 11.2.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nPerform the following operations on \\(f(x) = 2x^2 - 4\\) and \\(g(x) = x^2 + 4x - 2\\).\r\n
    \r\n \t
  1. \\(f(x) + g(x)\\)Addition yields \\(2x^2 - 4 + x^2 + 4x - 2\\), which simplifies to \\(3x^2 + 4x - 6\\).<\/li>\r\n \t
  2. \\(f(x) - g(x)\\)Subtraction yields \\(2x^2-4-(x^2+4x-2)\\), which simplifies to \\(x^2-4x-2\\).<\/li>\r\n \t
  3. \\(f(x)\\cdot g(x)\\)Multiplication yields \\((2x^2-4)(x^2+4x-2)\\), which simplifies to \\(2x^4+8x^3-4x^2-16x+8\\).<\/li>\r\n \t
  4. \\(f(x)\\div g(x)\\)Division yields \\((2x^2-4)\\div (x^2+4x-2)\\), which cannot be reduced any further.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nOften, you are asked to evaluate operations on functions where you must substitute some given value into the combined functions. Consider the following.\r\n
    \r\n

    Example 11.2.2<\/p>\r\n\r\n<\/header>\r\n

    \r\n\r\nPerform the following operations on \\(f(x) = x^2 - 3\\) and \\(g(x) = 2x^2 + 3x\\) and evaluate for the given values.\r\n
      \r\n \t
    1. \\(f(2) + g(2)\\)\r\n\\([x^2-3]+[2x^2+3x]\\)\r\n\\([(2)^2-3]+[2(2)^2+3(2)]\\)\r\n\\(4-3+8+6=15\\)\r\n\\(f(2)+g(2)=15\\)<\/li>\r\n \t
    2. \\(f(1) - g(3)\\)\r\n\\([x^2-3]-[2x^2+3x]\\)\r\n\\([(1)^2-3]-[2(3)^2+3(3)]\\)\r\n\\([1-3]-[18+9]=-29\\)\r\n\\(f(1)-g(3)=-29\\)<\/li>\r\n \t
    3. \\(f(0)\\cdot g(2)\\)\r\n\\([x^2-3]\\cdot [2x^2+3x]\\)\r\n\\([0^2-3]\\cdot [2(2)^2+3(2)]\\)\r\n\\([-3]\\cdot [8+6]=-42\\)\r\n\\(f(0)\\cdot g(2)=-42\\)<\/li>\r\n \t
    4. \\(f(2)\\div g(0)\\)\r\n\\([x^2-3]\\div [2x^2+3x]\\)\r\n\\([2^2-3]\\div [2(0)^2+3(0)]\\)\r\n\\([1]\\div [0]=\\text{ undefined}\\)<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nComposite functions are functions that involve substitution of functions, such as \\(f(x)\\) is substituted for the \\(x\\)-value in the \\(g(x)\\) function or the reverse. Which goes where is outlined by the way the equation is written:\r\n

      \\(\\begin{array}{l}\r\n(f \\circ g)(x)\\text{ means that the }g(x)\\text{ function is used to replace the }x\\text{-values in the }f(x)\\text{ function} \\\\\r\n(g\\circ f)(x)\\text{ means that the }f(x)\\text{ function is used to replace the }x\\text{-values in the }g(x)\\text{ function}\r\n\\end{array}\\)<\/p>\r\nThe more conventional way to write these composite functions is:\r\n\r\n\\[(f\\circ g)(x) = f(g(x))\\text{ and }(g\\circ f)(x) = g(f(x))\\]\r\n\r\nConsider the following examples of composite functions.\r\n

      \r\n

      Example 11.2.3<\/p>\r\n\r\n<\/header>\r\n

      \r\n\r\nGiven the functions \\(f(x) = 3x - 5\\) and \\(g(x) = x^2 + 2\\), evaluate for:\r\n
        \r\n \t
      1. \\((f\\circ g)(2)\\)\\(\\begin{array}{rrl}\r\n(f\\circ g)(x)&=&f(g(x)) \\\\\r\nf(g(x))&=&3(x^2+2)-5 \\\\\r\nf(g(2))&=&3(2^2+2)-5 \\\\\r\nf(g(2))&=&3(6)-5=13\r\n\\end{array}\\)<\/li>\r\n \t
      2. \\((g\\circ f)(-1)\\)\\(\\begin{array}{rrl}\r\n(g\\circ f)(x)&=&g(f(x)) \\\\\r\ng(f(x))&=&[3x-5]^2+2 \\\\\r\ng(f(-1))&=&[3(-1)-5]^2+2 \\\\\r\ng(f(-1))&=&[-8]^2+2 \\\\\r\ng(f(-1))&=&66\r\n\\end{array}\\)<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

        Questions<\/h1>\r\nPerform the indicated operations.\r\n
          \r\n \t
        1. \\(g(a) = a^3 + 5a^2\\)\r\n\\(f(a) = 2a + 4\\)\r\nFind \\(g(3) + f(3)\\)<\/li>\r\n \t
        2. \\(f(x) = -3x^2 + 3x\\)\r\n\\(g(x) = 2x + 5\\)\r\nFind \\(\\dfrac{f(-4)}{g(-4)}\\)<\/li>\r\n \t
        3. \\(g(x) = -4x + 1\\)\r\n\\(h(x) = -2x - 1\\)\r\nFind \\(g(5) + h(5)\\)<\/li>\r\n \t
        4. \\(g(x) = 3x + 1\\)\r\n\\(f(x) = x^3 + 3x^2\\)\r\nFind \\(g(2)\\cdot f(2)\\)<\/li>\r\n \t
        5. \\(g(t) = t - 3\\)\r\n\\(h(t) = -3t^3 + 6t\\)\r\nFind \\(g(1) + h(1)\\)<\/li>\r\n \t
        6. \\(g(x) = x^2 - 2\\)\r\n\\(h(x) = 2x + 5\\)\r\nFind \\(g(-6) + h(-6)\\)<\/li>\r\n \t
        7. \\(h(n) = 2n - 1\\)\r\n\\(g(n) = 3n - 5\\)\r\nFind \\(\\dfrac{h(0)}{g(0)}\\)<\/li>\r\n \t
        8. \\(g(a) = 3a - 2\\)\r\n\\(h(a) = 4a - 2\\)\r\nFind \\((g + h)(10)\\)<\/li>\r\n \t
        9. \\(g(a) = 3a + 3\\)\r\n\\(f(a) = 2a - 2\\)\r\nFind \\((g + f)(9)\\)<\/li>\r\n \t
        10. \\(g(x) = 4x + 3\\)\r\n\\(h(x) = x^3 - 2x^2\\)\r\nFind \\((g - h)(-1)\\)<\/li>\r\n \t
        11. \\(g(x) = x + 3\\)\r\n\\(f(x) = -x + 4\\)\r\nFind \\((g - f)(3)\\)<\/li>\r\n \t
        12. \\(g(x) = x^2 + 2\\)\r\n\\(f(x) = 2x + 5\\)\r\nFind \\((g - f)(0)\\)<\/li>\r\n \t
        13. \\(f(n) = n - 5\\)\r\n\\(g(n) = 4n + 2\\)\r\nFind \\((f + g)(-8)\\)<\/li>\r\n \t
        14. \\(h(t) = t + 5\\)\r\n\\(g(t) = 3t - 5\\)\r\nFind \\((h\\cdot g)(5)\\)<\/li>\r\n \t
        15. \\(g(t) = t - 4\\)\r\n\\(h(t) = 2t\\)\r\nFind \\((g\\cdot h)(3t)\\)<\/li>\r\n \t
        16. \\(g(n) = n^2 + 5\\)\r\n\\(f(n) = 3n + 5\\)\r\nFind \\(\\dfrac{g(n)}{f(n)}\\)<\/li>\r\n \t
        17. \\(g(a) = -2a + 5\\)\r\n\\(f(a) = 3a + 5\\)\r\nFind \\(\\left(\\dfrac{g}{f}\\right)(a^2)\\)<\/li>\r\n \t
        18. \\(h(n) = n^3 + 4n\\)\r\n\\(g(n) = 4n + 5\\)\r\nFind \\(h(n) + g(n)\\)<\/li>\r\n \t
        19. \\(g(n) = n^2 - 4n\\)\r\n\\(h(n) = n - 5\\)\r\nFind \\(g(n^2)\\cdot h(n^2)\\)<\/li>\r\n \t
        20. \\(g(n) = n + 5\\)\r\n\\(h(n) = 2n - 5\\)\r\nFind \\((g\\cdot h)(-3n)\\)<\/li>\r\n<\/ol>\r\nSolve the following composite functions.\r\n
            \r\n \t
          1. \\(f(x) = -4x + 1\\)\r\n\\(g(x) = 4x + 3\\)\r\nFind \\((f\\circ g)(9)\\)<\/li>\r\n \t
          2. \\(h(a) = 3a + 3\\)\r\n\\(g(a) = a + 1\\)\r\nFind \\((h \\circ g)(5)\\)<\/li>\r\n \t
          3. \\(g(x) = x + 4\\)\r\n\\(h(x) = x^2 - 1\\)\r\nFind \\((g \\circ h)(10)\\)<\/li>\r\n \t
          4. \\(f(n) = -4n + 2\\)\r\n\\(g(n) = n + 4\\)\r\nFind \\((f \\circ g)(9)\\)<\/li>\r\n \t
          5. \\(g(x) = 2x - 4\\)\r\n\\(h(x) = 2x^3 + 4x^2\\)\r\nFind \\((g \\circ h)(3)\\)<\/li>\r\n \t
          6. \\(g(x) = x^2 - 5x\\)\r\n\\(h(x) = 4x + 4\\)\r\nFind \\((g \\circ h)(x)\\)<\/li>\r\n \t
          7. \\(f(a) = -2a + 2\\)\r\n\\(g(a) = 4a\\)\r\nFind \\((f \\circ g)(a)\\)<\/li>\r\n \t
          8. \\(g(x) = 4x + 4\\)\r\n\\(f(x) = x^3 - 1\\)\r\nFind \\((g \\circ f)(x)\\)<\/li>\r\n \t
          9. \\(g(x) = -x + 5\\)\r\n\\(f(x) = 2x - 3\\)\r\nFind \\((g \\circ f)(x)\\)<\/li>\r\n \t
          10. \\(f(t) = 4t + 3\\)\r\n\\(g(t) = -4t - 2\\)\r\nFind \\((f \\circ g)(t)\\)<\/li>\r\n<\/ol>\r\nAnswer Key 11.2<\/a>","rendered":"

            In Chapter 5, you solved systems of linear equations through substitution, addition, subtraction, multiplication, and division. A similar process is employed in this topic, where you will add, subtract, multiply, divide, or substitute functions. The notation used for this looks like the following:<\/p>\n

            Given two functions \"f(x)\" and \"g(x)\":<\/p>\n

            \"\begin{array}{clcl}<\/p>\n

            When encountering questions about operations on functions, you will generally be asked to do two things: combine the equations in some described fashion and to substitute some value to replace the variable in the original equation. These are illustrated in the following examples.<\/p>\n

            \n
            \n

            Example 11.2.1<\/p>\n<\/header>\n

            \n

            Perform the following operations on \"f(x) = 2x^2 - 4\" and \"g(x) = x^2 + 4x - 2\".<\/p>\n

              \n
            1. \"f(x) + g(x)\"Addition yields \"2x^2 - 4 + x^2 + 4x - 2\", which simplifies to \"3x^2 + 4x - 6\".<\/li>\n
            2. \"f(x) - g(x)\"Subtraction yields \"2x^2-4-(x^2+4x-2)\", which simplifies to \"x^2-4x-2\".<\/li>\n
            3. \"f(x)\cdot g(x)\"Multiplication yields \"(2x^2-4)(x^2+4x-2)\", which simplifies to \"2x^4+8x^3-4x^2-16x+8\".<\/li>\n
            4. \"f(x)\div g(x)\"Division yields \"(2x^2-4)\div (x^2+4x-2)\", which cannot be reduced any further.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

              Often, you are asked to evaluate operations on functions where you must substitute some given value into the combined functions. Consider the following.<\/p>\n

              \n
              \n

              Example 11.2.2<\/p>\n<\/header>\n

              \n

              Perform the following operations on \"f(x) = x^2 - 3\" and \"g(x) = 2x^2 + 3x\" and evaluate for the given values.<\/p>\n

                \n
              1. \"f(2) + g(2)\"
                \n\"[x^2-3]+[2x^2+3x]\"
                \n\"[(2)^2-3]+[2(2)^2+3(2)]\"
                \n\"4-3+8+6=15\"
                \n\"f(2)+g(2)=15\"<\/li>\n
              2. \"f(1) - g(3)\"
                \n\"[x^2-3]-[2x^2+3x]\"
                \n\"[(1)^2-3]-[2(3)^2+3(3)]\"
                \n\"[1-3]-[18+9]=-29\"
                \n\"f(1)-g(3)=-29\"<\/li>\n
              3. \"f(0)\cdot g(2)\"
                \n\"[x^2-3]\cdot [2x^2+3x]\"
                \n\"[0^2-3]\cdot [2(2)^2+3(2)]\"
                \n\"[-3]\cdot [8+6]=-42\"
                \n\"f(0)\cdot g(2)=-42\"<\/li>\n
              4. \"f(2)\div g(0)\"
                \n\"[x^2-3]\div [2x^2+3x]\"
                \n\"[2^2-3]\div [2(0)^2+3(0)]\"
                \n\"[1]\div [0]=\text{ undefined}\"<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                Composite functions are functions that involve substitution of functions, such as \"f(x)\" is substituted for the \"x\"-value in the \"g(x)\" function or the reverse. Which goes where is outlined by the way the equation is written:<\/p>\n

                \"\begin{array}{l}<\/p>\n

                The more conventional way to write these composite functions is:<\/p>\n

                  <\/span>   <\/span>\"\[(f\circ g)(x) = f(g(x))\text{ and }(g\circ f)(x) = g(f(x))\]\"<\/p>\n

                Consider the following examples of composite functions.<\/p>\n

                \n
                \n

                Example 11.2.3<\/p>\n<\/header>\n

                \n

                Given the functions \"f(x) = 3x - 5\" and \"g(x) = x^2 + 2\", evaluate for:<\/p>\n

                  \n
                1. \"(f\circ g)(2)\"\"\begin{array}{rrl}<\/li>\n
                2. \"(g\circ f)(-1)\"\"\begin{array}{rrl}<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                  Questions<\/h1>\n

                  Perform the indicated operations.<\/p>\n

                    \n
                  1. \"g(a) = a^3 + 5a^2\"
                    \n\"f(a) = 2a + 4\"
                    \nFind \"g(3) + f(3)\"<\/li>\n
                  2. \"f(x) = -3x^2 + 3x\"
                    \n\"g(x) = 2x + 5\"
                    \nFind \"\dfrac{f(-4)}{g(-4)}\"<\/li>\n
                  3. \"g(x) = -4x + 1\"
                    \n\"h(x) = -2x - 1\"
                    \nFind \"g(5) + h(5)\"<\/li>\n
                  4. \"g(x) = 3x + 1\"
                    \n\"f(x) = x^3 + 3x^2\"
                    \nFind \"g(2)\cdot f(2)\"<\/li>\n
                  5. \"g(t) = t - 3\"
                    \n\"h(t) = -3t^3 + 6t\"
                    \nFind \"g(1) + h(1)\"<\/li>\n
                  6. \"g(x) = x^2 - 2\"
                    \n\"h(x) = 2x + 5\"
                    \nFind \"g(-6) + h(-6)\"<\/li>\n
                  7. \"h(n) = 2n - 1\"
                    \n\"g(n) = 3n - 5\"
                    \nFind \"\dfrac{h(0)}{g(0)}\"<\/li>\n
                  8. \"g(a) = 3a - 2\"
                    \n\"h(a) = 4a - 2\"
                    \nFind \"(g + h)(10)\"<\/li>\n
                  9. \"g(a) = 3a + 3\"
                    \n\"f(a) = 2a - 2\"
                    \nFind \"(g + f)(9)\"<\/li>\n
                  10. \"g(x) = 4x + 3\"
                    \n\"h(x) = x^3 - 2x^2\"
                    \nFind \"(g - h)(-1)\"<\/li>\n
                  11. \"g(x) = x + 3\"
                    \n\"f(x) = -x + 4\"
                    \nFind \"(g - f)(3)\"<\/li>\n
                  12. \"g(x) = x^2 + 2\"
                    \n\"f(x) = 2x + 5\"
                    \nFind \"(g - f)(0)\"<\/li>\n
                  13. \"f(n) = n - 5\"
                    \n\"g(n) = 4n + 2\"
                    \nFind \"(f + g)(-8)\"<\/li>\n
                  14. \"h(t) = t + 5\"
                    \n\"g(t) = 3t - 5\"
                    \nFind \"(h\cdot g)(5)\"<\/li>\n
                  15. \"g(t) = t - 4\"
                    \n\"h(t) = 2t\"
                    \nFind \"(g\cdot h)(3t)\"<\/li>\n
                  16. \"g(n) = n^2 + 5\"
                    \n\"f(n) = 3n + 5\"
                    \nFind \"\dfrac{g(n)}{f(n)}\"<\/li>\n
                  17. \"g(a) = -2a + 5\"
                    \n\"f(a) = 3a + 5\"
                    \nFind \"\left(\dfrac{g}{f}\right)(a^2)\"<\/li>\n
                  18. \"h(n) = n^3 + 4n\"
                    \n\"g(n) = 4n + 5\"
                    \nFind \"h(n) + g(n)\"<\/li>\n
                  19. \"g(n) = n^2 - 4n\"
                    \n\"h(n) = n - 5\"
                    \nFind \"g(n^2)\cdot h(n^2)\"<\/li>\n
                  20. \"g(n) = n + 5\"
                    \n\"h(n) = 2n - 5\"
                    \nFind \"(g\cdot h)(-3n)\"<\/li>\n<\/ol>\n

                    Solve the following composite functions.<\/p>\n

                      \n
                    1. \"f(x) = -4x + 1\"
                      \n\"g(x) = 4x + 3\"
                      \nFind \"(f\circ g)(9)\"<\/li>\n
                    2. \"h(a) = 3a + 3\"
                      \n\"g(a) = a + 1\"
                      \nFind \"(h \circ g)(5)\"<\/li>\n
                    3. \"g(x) = x + 4\"
                      \n\"h(x) = x^2 - 1\"
                      \nFind \"(g \circ h)(10)\"<\/li>\n
                    4. \"f(n) = -4n + 2\"
                      \n\"g(n) = n + 4\"
                      \nFind \"(f \circ g)(9)\"<\/li>\n
                    5. \"g(x) = 2x - 4\"
                      \n\"h(x) = 2x^3 + 4x^2\"
                      \nFind \"(g \circ h)(3)\"<\/li>\n
                    6. \"g(x) = x^2 - 5x\"
                      \n\"h(x) = 4x + 4\"
                      \nFind \"(g \circ h)(x)\"<\/li>\n
                    7. \"f(a) = -2a + 2\"
                      \n\"g(a) = 4a\"
                      \nFind \"(f \circ g)(a)\"<\/li>\n
                    8. \"g(x) = 4x + 4\"
                      \n\"f(x) = x^3 - 1\"
                      \nFind \"(g \circ f)(x)\"<\/li>\n
                    9. \"g(x) = -x + 5\"
                      \n\"f(x) = 2x - 3\"
                      \nFind \"(g \circ f)(x)\"<\/li>\n
                    10. \"f(t) = 4t + 3\"
                      \n\"g(t) = -4t - 2\"
                      \nFind \"(f \circ g)(t)\"<\/li>\n<\/ol>\n

                      Answer Key 11.2<\/a><\/p>\n","protected":false},"author":540,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-773","chapter","type-chapter","status-publish","hentry"],"part":399,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/773","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/users\/540"}],"version-history":[{"count":9,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/773\/revisions"}],"predecessor-version":[{"id":3664,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/773\/revisions\/3664"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/parts\/399"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/773\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=773"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapter-type?post=773"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=773"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=773"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}