{"id":2069,"date":"2025-08-04T15:50:43","date_gmt":"2025-08-04T15:50:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2069"},"modified":"2025-12-29T17:15:55","modified_gmt":"2025-12-29T17:15:55","slug":"working-with-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/working-with-functions-get-stronger\/","title":{"raw":"Working with Functions: Get Stronger","rendered":"Working with Functions: Get Stronger"},"content":{"raw":"
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Composition of Functions<\/h2>\r\n1. How does one find the domain of the quotient of two functions, [latex]\\frac{f}{g}?[\/latex]\r\n\r\n3. If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.\r\n\r\n7. Given [latex]f\\left(x\\right)=2{x}^{2}+4x\\text{ }[\/latex] and [latex]\\text{ }g\\left(x\\right)=\\frac{1}{2x}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.\r\n\r\n[reveal-answer q=\"568911\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"568911\"]When finding the domain of [latex]\\frac{f}{g}[\/latex], remember two restrictions: (1) the denominator [latex]g(x)[\/latex] cannot equal zero, and (2) both [latex]f[\/latex] and [latex]g[\/latex] must be defined. Start by finding where [latex]g(x) = 0[\/latex].[\/hidden-answer]\r\n\r\n11. Given [latex]f\\left(x\\right)=2{x}^{2}+1[\/latex] and [latex]g\\left(x\\right)=3x - 5[\/latex], find the following:\r\n

[latex]f\\left(g\\left(2\\right)\\right)[\/latex]\r\n[latex]f\\left(g\\left(x\\right)\\right)[\/latex]\r\n[latex]g\\left(f\\left(x\\right)\\right)[\/latex]\r\n[latex]\\left(g\\circ g\\right)\\left(x\\right)[\/latex]\r\n[latex]\\left(f\\circ f\\right)\\left(-2\\right)[\/latex]<\/p>\r\nFor the following exercises, use each pair of functions to find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex]. Simplify your answers.\r\n\r\n13. [latex]f\\left(x\\right)=\\sqrt{x}+2,g\\left(x\\right)={x}^{2}+3[\/latex]\r\n\r\n25. For [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] and [latex]g\\left(x\\right)=\\sqrt{x - 1}[\/latex], write the domain of [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex] in interval notation.\r\n\r\n[reveal-answer q=\"4138\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"4138\"] For the domain of [latex](f \\circ g)(x) = f(g(x))[\/latex], you need: (1) [latex]x[\/latex] values where [latex]g(x)[\/latex] is defined, AND (2) the output of [latex]g(x)[\/latex] must be in the domain of [latex]f[\/latex]. Set up both conditions as inequalities.[\/hidden-answer]\r\n\r\nFor the following exercises, find functions [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] so the given function can be expressed as [latex]h\\left(x\\right)=f\\left(g\\left(x\\right)\\right)[\/latex].\r\n\r\n27. [latex]h\\left(x\\right)={\\left(x - 5\\right)}^{3}[\/latex]\r\n\r\n35. [latex]h\\left(x\\right)=\\sqrt{2x+6}[\/latex]\r\n\r\nFor the following exercises, use the graphs of [latex]f[\/latex]\u00a0and [latex]g[\/latex]\u00a0to evaluate the expressions.\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]\"Graph f(x)[\/caption]\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]\"Graph g(x)[\/caption]\r\n\r\n

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43. [latex]f\\left(g\\left(1\\right)\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"862714\"]Hint [\/reveal-answer]\r\n[hidden-answer a=\"862714\"]Work from the inside out. For [latex]f(g(1))[\/latex], first find [latex]g(1)[\/latex] on the graph of [latex]g[\/latex], then use that output as the input for [latex]f[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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45. [latex]g\\left(f\\left(0\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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For the following exercises, use the function values for [latex]f\\text{ and }g[\/latex]\u00a0to evaluate each expression.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n
0<\/td>\r\n7<\/td>\r\n9<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n6<\/td>\r\n5<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n5<\/td>\r\n6<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n8<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n4<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n0<\/td>\r\n8<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n2<\/td>\r\n7<\/td>\r\n<\/tr>\r\n
7<\/td>\r\n1<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
8<\/td>\r\n9<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
9<\/td>\r\n3<\/td>\r\n0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
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59. [latex]f\\left(g\\left(5\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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61. [latex]g\\left(f\\left(3\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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63. [latex]f\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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For the following exercises, use each pair of functions to find [latex]f\\left(g\\left(0\\right)\\right)[\/latex] and [latex]g\\left(f\\left(0\\right)\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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73. [latex]f\\left(x\\right)=5x+7,g\\left(x\\right)=4 - 2{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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91. The function [latex]A\\left(d\\right)[\/latex] gives the pain level on a scale of 0 to 10 experienced by a patient with [latex]d[\/latex] milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient\u2019s system after [latex]t[\/latex] minutes is modeled by [latex]m\\left(t\\right)[\/latex]. Which of the following would you do in order to determine when the patient will be at a pain level of 4?<\/p>\r\n

a. Evaluate [latex]A\\left(m\\left(4\\right)\\right)[\/latex].<\/p>\r\n

b. Evaluate [latex]m\\left(A\\left(4\\right)\\right)[\/latex].<\/p>\r\n

c. Solve [latex]A\\left(m\\left(t\\right)\\right)=4[\/latex].<\/p>\r\n

d. Solve [latex]m\\left(A\\left(d\\right)\\right)=4[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Transformation of Functions<\/h2>\r\n1. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?\r\n\r\n3. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?\r\n\r\n7. Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=|x|[\/latex]\r\nis shifted down 3 units and to the right 1 unit.\r\n\r\nFor the following exercises, describe how the graph of the function is a transformation of the graph of the original function [latex]f[\/latex].\r\n\r\n11. [latex]y=f\\left(x+43\\right)[\/latex]\r\n\r\n15. [latex]y=f\\left(x\\right)+8[\/latex]\r\n\r\n19. [latex]y=f\\left(x+4\\right)-1[\/latex]\r\n\r\nFor the following exercises, determine the interval(s) on which the function is increasing and decreasing.\r\n\r\n21. [latex]g\\left(x\\right)=5{\\left(x+3\\right)}^{2}-2[\/latex]\r\n\r\n23. [latex]k\\left(x\\right)=-3\\sqrt{x}-1[\/latex]\r\n\r\nFor the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.\r\n\r\n27. [latex]f\\left(t\\right)={\\left(t+1\\right)}^{2}-3[\/latex]\r\n\r\n29. [latex]k\\left(x\\right)={\\left(x - 2\\right)}^{3}-1[\/latex]\r\n\r\n31.\u00a0Tabular representations for the functions [latex]f,g[\/latex], and [latex]h[\/latex] are given below. Write [latex]g\\left(x\\right)[\/latex] and [latex]h\\left(x\\right)[\/latex] as transformations of [latex]f\\left(x\\right)[\/latex].\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n\u22123<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
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[latex]x[\/latex]<\/strong><\/td>\r\n\u22121<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n\u22123<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\r\n\u22121<\/td>\r\n0<\/td>\r\n\u22122<\/td>\r\n2<\/td>\r\n3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
<\/div>\r\n<\/div>\r\n

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.<\/p>\r\n\r\n

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\r\n\r\n33.\r\n\"Graph<\/span>\r\n\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n \r\n
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For the following exercises, determine whether the function is odd, even, or neither.<\/p>\r\n\r\n

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47. [latex]f\\left(x\\right)=3{x}^{4}[\/latex]<\/p>\r\n[reveal-answer q=\"405044\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"405044\"]Test [latex]f(-x)[\/latex] and compare it to [latex]f(x)[\/latex]. If [latex]f(-x) = f(x)[\/latex], it's even. If [latex]f(-x) = -f(x)[\/latex], it's odd.[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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49. [latex]h\\left(x\\right)=\\frac{1}{x}+3x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function [latex]f[\/latex].<\/p>\r\n\r\n

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53. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57. [latex]g\\left(x\\right)=f\\left(5x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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59. [latex]g\\left(x\\right)=f\\left(\\frac{1}{3}x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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61. [latex]g\\left(x\\right)=3f\\left(-x\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"551459\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"551459\"]Work from inside to outside: (1) [latex]-x[\/latex] and (2) the 3 outside.[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, write a formula for the function [latex]g[\/latex] that results when the graph of a given toolkit function is transformed as described.<\/p>\r\n\r\n

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65. The graph of [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex] is vertically compressed by a factor of [latex]\\frac{1}{3}[\/latex], then shifted to the left 2 units and down 3 units.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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67. The graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] is vertically compressed by a factor of [latex]\\frac{1}{2}[\/latex], then shifted to the right 5 units and up 1 unit.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.<\/p>\r\n\r\n

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69. [latex]g\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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77. [latex]a\\left(x\\right)=\\sqrt{-x+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Inverse Functions<\/h2>\r\n1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?\r\n\r\n3. Can a function be its own inverse? Explain.\r\n\r\n5. How do you find the inverse of a function algebraically?\r\n\r\nFor the following exercises, find [latex]{f}^{-1}\\left(x\\right)[\/latex] for each function.\r\n\r\n7. [latex]f\\left(x\\right)=x+3[\/latex]\r\n\r\n[reveal-answer q=\"383417\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"383417\"]The standard steps are: (1) Replace [latex]f(x)[\/latex] with [latex]y[\/latex], (2) Swap [latex]x[\/latex] and [latex]y[\/latex], (3) Solve for [latex]y[\/latex], (4) Replace [latex]y[\/latex] with [latex]f^{-1}(x)[\/latex].[\/hidden-answer]\r\n\r\n11.\u00a0[latex]f\\left(x\\right)=\\frac{x}{x+2}[\/latex]\r\n\r\nFor the following exercises, find a domain on which each function [latex]f[\/latex] is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of [latex]f[\/latex] restricted to that domain.\r\n\r\n13. [latex]f\\left(x\\right)={\\left(x+7\\right)}^{2}[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)={x}^{2}-5[\/latex]\r\n\r\nFor the following exercises, use function composition to verify that [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] are inverse functions.\r\n\r\n17. [latex]f\\left(x\\right)=\\sqrt[3]{x - 1}[\/latex] and [latex]g\\left(x\\right)={x}^{3}+1[\/latex]\r\n\r\nFor the following exercises, use the graph of [latex]f[\/latex] shown below.\r\n\"Graph\r\n25. Find [latex]f\\left(0\\right)[\/latex].\r\n\r\n27. Find [latex]{f}^{-1}\\left(0\\right)[\/latex].\r\n\r\nFor the following exercises, use the graph of the one-to-one function shown below.\r\n\"Graph\r\n29. Sketch the graph of [latex]{f}^{-1}[\/latex].\r\n\r\nFor the following exercises, evaluate or solve, assuming that the function [latex]f[\/latex] is one-to-one.\r\n\r\n33. If [latex]f\\left(6\\right)=7[\/latex], find [latex]{f}^{-1}\\left(7\\right)[\/latex].\r\n\r\n35. If [latex]{f}^{-1}\\left(-4\\right)=-8[\/latex], find [latex]f\\left(-8\\right)[\/latex].\r\n\r\nFor the following exercises, use the values listed in the table below\u00a0to evaluate or solve.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n
0<\/td>\r\n8<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n7<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n6<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n5<\/td>\r\n<\/tr>\r\n
7<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
8<\/td>\r\n9<\/td>\r\n<\/tr>\r\n
9<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n37. Find [latex]f\\left(1\\right)[\/latex].\r\n\r\n39. Find [latex]{f}^{-1}\\left(0\\right)[\/latex].\r\n\r\n41. Use the tabular representation of [latex]f[\/latex] to create a table for [latex]{f}^{-1}\\left(x\\right)[\/latex].\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n3<\/td>\r\n6<\/td>\r\n9<\/td>\r\n13<\/td>\r\n14<\/td>\r\n<\/tr>\r\n
[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n1<\/td>\r\n4<\/td>\r\n7<\/td>\r\n12<\/td>\r\n16<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"621873\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"621873\"]To create the inverse table, swap the [latex]x[\/latex] and [latex]f(x)[\/latex] rows. The outputs of [latex]f[\/latex] become the inputs of [latex]f^{-1}[\/latex], and vice versa.[\/hidden-answer]\r\n\r\n45.\u00a0To convert from [latex]x[\/latex] degrees Celsius to [latex]y[\/latex] degrees Fahrenheit, we use the formula [latex]f\\left(x\\right)=\\frac{9}{5}x+32[\/latex]. Find the inverse function, if it exists, and explain its meaning.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"
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Composition of Functions<\/h2>\n

1. How does one find the domain of the quotient of two functions, [latex]\\frac{f}{g}?[\/latex]<\/p>\n

3. If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.<\/p>\n

7. Given [latex]f\\left(x\\right)=2{x}^{2}+4x\\text{ }[\/latex] and [latex]\\text{ }g\\left(x\\right)=\\frac{1}{2x}[\/latex], find [latex]f+g,f-g,fg,\\text{ }[\/latex] and [latex]\\text{ }\\frac{f}{g}[\/latex]. Determine the domain for each function in interval notation.<\/p>\n