{"id":1505,"date":"2021-12-02T19:38:19","date_gmt":"2021-12-03T00:38:19","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/11-2-operations-on-functions\/"},"modified":"2023-08-31T19:24:00","modified_gmt":"2023-08-31T23:24:00","slug":"operations-on-functions","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/operations-on-functions\/","title":{"raw":"11.2 Operations on Functions","rendered":"11.2 Operations on Functions"},"content":{"raw":"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:\r\n\r\nGiven two functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex]:\r\n

[latex]\\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}[\/latex]<\/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 [latex]f(x) = 2x^2 - 4[\/latex] and [latex]g(x) = x^2 + 4x - 2[\/latex].\r\n
    \r\n \t
  1. [latex]f(x) + g(x)[\/latex]Addition yields [latex]2x^2 - 4 + x^2 + 4x - 2[\/latex], which simplifies to [latex]3x^2 + 4x - 6[\/latex].<\/li>\r\n \t
  2. [latex]f(x) - g(x)[\/latex]Subtraction yields [latex]2x^2-4-(x^2+4x-2)[\/latex], which simplifies to [latex]x^2-4x-2[\/latex].<\/li>\r\n \t
  3. [latex]f(x)\\cdot g(x)[\/latex]Multiplication yields [latex](2x^2-4)(x^2+4x-2)[\/latex], which simplifies to [latex]2x^4+8x^3-4x^2-16x+8[\/latex].<\/li>\r\n \t
  4. [latex]f(x)\\div g(x)[\/latex]Division yields [latex](2x^2-4)\\div (x^2+4x-2)[\/latex], 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 [latex]f(x) = x^2 - 3[\/latex] and [latex]g(x) = 2x^2 + 3x[\/latex] and evaluate for the given values.\r\n
      \r\n \t
    1. [latex]f(2) + g(2)[\/latex]\r\n[latex][x^2-3]+[2x^2+3x][\/latex]\r\n[latex][(2)^2-3]+[2(2)^2+3(2)][\/latex]\r\n[latex]4-3+8+6=15[\/latex]\r\n[latex]f(2)+g(2)=15[\/latex]<\/li>\r\n \t
    2. [latex]f(1) - g(3)[\/latex]\r\n[latex][x^2-3]-[2x^2+3x][\/latex]\r\n[latex][(1)^2-3]-[2(3)^2+3(3)][\/latex]\r\n[latex][1-3]-[18+9]=-29[\/latex]\r\n[latex]f(1)-g(3)=-29[\/latex]<\/li>\r\n \t
    3. [latex]f(0)\\cdot g(2)[\/latex]\r\n[latex][x^2-3]\\cdot [2x^2+3x][\/latex]\r\n[latex][0^2-3]\\cdot [2(2)^2+3(2)][\/latex]\r\n[latex][-3]\\cdot [8+6]=-42[\/latex]\r\n[latex]f(0)\\cdot g(2)=-42[\/latex]<\/li>\r\n \t
    4. [latex]f(2)\\div g(0)[\/latex]\r\n[latex][x^2-3]\\div [2x^2+3x][\/latex]\r\n[latex][2^2-3]\\div [2(0)^2+3(0)][\/latex]\r\n[latex][1]\\div [0]=\\text{ undefined}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nComposite functions are functions that involve substitution of functions, such as [latex]f(x)[\/latex] is substituted for the [latex]x[\/latex]-value in the [latex]g(x)[\/latex] function or the reverse. Which goes where is outlined by the way the equation is written:\r\n

      [latex]\\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}[\/latex]<\/p>\r\nThe more conventional way to write these composite functions is:\r\n\r\n[latex](f\\circ g)(x) = f(g(x))\\text{ and }(g\\circ f)(x) = g(f(x))[\/latex]\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 [latex]f(x) = 3x - 5[\/latex] and [latex]g(x) = x^2 + 2[\/latex], evaluate for:\r\n
        \r\n \t
      1. [latex](f\\circ g)(2)[\/latex][latex]\\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}[\/latex]<\/li>\r\n \t
      2. [latex](g\\circ f)(-1)[\/latex][latex]\\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}[\/latex]<\/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. [latex]g(a) = a^3 + 5a^2[\/latex]\r\n[latex]f(a) = 2a + 4[\/latex]\r\nFind [latex]g(3) + f(3)[\/latex]<\/li>\r\n \t
        2. [latex]f(x) = -3x^2 + 3x[\/latex]\r\n[latex]g(x) = 2x + 5[\/latex]\r\nFind [latex]\\dfrac{f(-4)}{g(-4)}[\/latex]<\/li>\r\n \t
        3. [latex]g(x) = -4x + 1[\/latex]\r\n[latex]h(x) = -2x - 1[\/latex]\r\nFind [latex]g(5) + h(5)[\/latex]<\/li>\r\n \t
        4. [latex]g(x) = 3x + 1[\/latex]\r\n[latex]f(x) = x^3 + 3x^2[\/latex]\r\nFind [latex]g(2)\\cdot f(2)[\/latex]<\/li>\r\n \t
        5. [latex]g(t) = t - 3[\/latex]\r\n[latex]h(t) = -3t^3 + 6t[\/latex]\r\nFind [latex]g(1) + h(1)[\/latex]<\/li>\r\n \t
        6. [latex]g(x) = x^2 - 2[\/latex]\r\n[latex]h(x) = 2x + 5[\/latex]\r\nFind [latex]g(-6) + h(-6)[\/latex]<\/li>\r\n \t
        7. [latex]h(n) = 2n - 1[\/latex]\r\n[latex]g(n) = 3n - 5[\/latex]\r\nFind [latex]\\dfrac{h(0)}{g(0)}[\/latex]<\/li>\r\n \t
        8. [latex]g(a) = 3a - 2[\/latex]\r\n[latex]h(a) = 4a - 2[\/latex]\r\nFind [latex](g + h)(10)[\/latex]<\/li>\r\n \t
        9. [latex]g(a) = 3a + 3[\/latex]\r\n[latex]f(a) = 2a - 2[\/latex]\r\nFind [latex](g + f)(9)[\/latex]<\/li>\r\n \t
        10. [latex]g(x) = 4x + 3[\/latex]\r\n[latex]h(x) = x^3 - 2x^2[\/latex]\r\nFind [latex](g - h)(-1)[\/latex]<\/li>\r\n \t
        11. [latex]g(x) = x + 3[\/latex]\r\n[latex]f(x) = -x + 4[\/latex]\r\nFind [latex](g - f)(3)[\/latex]<\/li>\r\n \t
        12. [latex]g(x) = x^2 + 2[\/latex]\r\n[latex]f(x) = 2x + 5[\/latex]\r\nFind [latex](g - f)(0)[\/latex]<\/li>\r\n \t
        13. [latex]f(n) = n - 5[\/latex]\r\n[latex]g(n) = 4n + 2[\/latex]\r\nFind [latex](f + g)(-8)[\/latex]<\/li>\r\n \t
        14. [latex]h(t) = t + 5[\/latex]\r\n[latex]g(t) = 3t - 5[\/latex]\r\nFind [latex](h\\cdot g)(5)[\/latex]<\/li>\r\n \t
        15. [latex]g(t) = t - 4[\/latex]\r\n[latex]h(t) = 2t[\/latex]\r\nFind [latex](g\\cdot h)(3t)[\/latex]<\/li>\r\n \t
        16. [latex]g(n) = n^2 + 5[\/latex]\r\n[latex]f(n) = 3n + 5[\/latex]\r\nFind [latex]\\dfrac{g(n)}{f(n)}[\/latex]<\/li>\r\n \t
        17. [latex]g(a) = -2a + 5[\/latex]\r\n[latex]f(a) = 3a + 5[\/latex]\r\nFind [latex]\\left(\\dfrac{g}{f}\\right)(a^2)[\/latex]<\/li>\r\n \t
        18. [latex]h(n) = n^3 + 4n[\/latex]\r\n[latex]g(n) = 4n + 5[\/latex]\r\nFind [latex]h(n) + g(n)[\/latex]<\/li>\r\n \t
        19. [latex]g(n) = n^2 - 4n[\/latex]\r\n[latex]h(n) = n - 5[\/latex]\r\nFind [latex]g(n^2)\\cdot h(n^2)[\/latex]<\/li>\r\n \t
        20. [latex]g(n) = n + 5[\/latex]\r\n[latex]h(n) = 2n - 5[\/latex]\r\nFind [latex](g\\cdot h)(-3n)[\/latex]<\/li>\r\n<\/ol>\r\nSolve the following composite functions.\r\n
            \r\n \t
          1. [latex]f(x) = -4x + 1[\/latex]\r\n[latex]g(x) = 4x + 3[\/latex]\r\nFind [latex](f\\circ g)(9)[\/latex]<\/li>\r\n \t
          2. [latex]h(a) = 3a + 3[\/latex]\r\n[latex]g(a) = a + 1[\/latex]\r\nFind [latex](h \\circ g)(5)[\/latex]<\/li>\r\n \t
          3. [latex]g(x) = x + 4[\/latex]\r\n[latex]h(x) = x^2 - 1[\/latex]\r\nFind [latex](g \\circ h)(10)[\/latex]<\/li>\r\n \t
          4. [latex]f(n) = -4n + 2[\/latex]\r\n[latex]g(n) = n + 4[\/latex]\r\nFind [latex](f \\circ g)(9)[\/latex]<\/li>\r\n \t
          5. [latex]g(x) = 2x - 4[\/latex]\r\n[latex]h(x) = 2x^3 + 4x^2[\/latex]\r\nFind [latex](g \\circ h)(3)[\/latex]<\/li>\r\n \t
          6. [latex]g(x) = x^2 - 5x[\/latex]\r\n[latex]h(x) = 4x + 4[\/latex]\r\nFind [latex](g \\circ h)(x)[\/latex]<\/li>\r\n \t
          7. [latex]f(a) = -2a + 2[\/latex]\r\n[latex]g(a) = 4a[\/latex]\r\nFind [latex](f \\circ g)(a)[\/latex]<\/li>\r\n \t
          8. [latex]g(x) = 4x + 4[\/latex]\r\n[latex]f(x) = x^3 - 1[\/latex]\r\nFind [latex](g \\circ f)(x)[\/latex]<\/li>\r\n \t
          9. [latex]g(x) = -x + 5[\/latex]\r\n[latex]f(x) = 2x - 3[\/latex]\r\nFind [latex](g \\circ f)(x)[\/latex]<\/li>\r\n \t
          10. [latex]f(t) = 4t + 3[\/latex]\r\n[latex]g(t) = -4t - 2[\/latex]\r\nFind [latex](f \\circ g)(t)[\/latex]<\/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 [latex]f(x)[\/latex] and [latex]g(x)[\/latex]:<\/p>\n

            [latex]\\begin{array}{clcl} f(x) + g(x)&\\text{ is the same as }&(f + g)(x)&\\text{ and means the addition of these two functions} \\\\ f(x) - g(x)&\\text{ is the same as }&(f - g)(x)&\\text{ and means the subtraction of these two functions} \\\\ f(x)\\cdot g(x)&\\text{ is the same as }&(f\\cdot g)(x)&\\text{ and means the multiplication of these two functions} \\\\ f(x)\\div g(x)&\\text{ is the same as }&(f\\div g)(x)&\\text{ and means the addition of these two functions} \\end{array}[\/latex]<\/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 [latex]f(x) = 2x^2 - 4[\/latex] and [latex]g(x) = x^2 + 4x - 2[\/latex].<\/p>\n

              \n
            1. [latex]f(x) + g(x)[\/latex]Addition yields [latex]2x^2 - 4 + x^2 + 4x - 2[\/latex], which simplifies to [latex]3x^2 + 4x - 6[\/latex].<\/li>\n
            2. [latex]f(x) - g(x)[\/latex]Subtraction yields [latex]2x^2-4-(x^2+4x-2)[\/latex], which simplifies to [latex]x^2-4x-2[\/latex].<\/li>\n
            3. [latex]f(x)\\cdot g(x)[\/latex]Multiplication yields [latex](2x^2-4)(x^2+4x-2)[\/latex], which simplifies to [latex]2x^4+8x^3-4x^2-16x+8[\/latex].<\/li>\n
            4. [latex]f(x)\\div g(x)[\/latex]Division yields [latex](2x^2-4)\\div (x^2+4x-2)[\/latex], 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 [latex]f(x) = x^2 - 3[\/latex] and [latex]g(x) = 2x^2 + 3x[\/latex] and evaluate for the given values.<\/p>\n

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

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

                [latex]\\begin{array}{l} (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} \\\\ (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} \\end{array}[\/latex]<\/p>\n

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

                [latex](f\\circ g)(x) = f(g(x))\\text{ and }(g\\circ f)(x) = g(f(x))[\/latex]<\/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 [latex]f(x) = 3x - 5[\/latex] and [latex]g(x) = x^2 + 2[\/latex], evaluate for:<\/p>\n

                  \n
                1. [latex](f\\circ g)(2)[\/latex][latex]\\begin{array}{rrl} (f\\circ g)(x)&=&f(g(x)) \\\\ f(g(x))&=&3(x^2+2)-5 \\\\ f(g(2))&=&3(2^2+2)-5 \\\\ f(g(2))&=&3(6)-5=13 \\end{array}[\/latex]<\/li>\n
                2. [latex](g\\circ f)(-1)[\/latex][latex]\\begin{array}{rrl} (g\\circ f)(x)&=&g(f(x)) \\\\ g(f(x))&=&[3x-5]^2+2 \\\\ g(f(-1))&=&[3(-1)-5]^2+2 \\\\ g(f(-1))&=&[-8]^2+2 \\\\ g(f(-1))&=&66 \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                  Questions<\/h1>\n

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

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

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

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

                      Answer Key 11.2<\/a><\/p>\n","protected":false},"author":90,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1505","chapter","type-chapter","status-publish","hentry","license-cc-by-nc-sa"],"part":1491,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1505","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":2,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1505\/revisions"}],"predecessor-version":[{"id":2173,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1505\/revisions\/2173"}],"part":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/parts\/1491"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1505\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1505"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapter-type?post=1505"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1505"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1505"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}