{"id":15513,"date":"2019-09-04T19:38:27","date_gmt":"2019-09-04T19:38:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&p=15513"},"modified":"2025-02-05T05:18:11","modified_gmt":"2025-02-05T05:18:11","slug":"problem-set-7-inverse-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-7-inverse-functions\/","title":{"raw":"Problem Set 7: Inverse Functions","rendered":"Problem Set 7: Inverse Functions"},"content":{"raw":"1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?\r\n\r\n2. Why do we restrict the domain of the function [latex]f\\left(x\\right)={x}^{2}[\/latex] to find the function\u2019s inverse?\r\n\r\n3. Can a function be its own inverse? Explain.\r\n\r\n4. Are one-to-one functions either always increasing or always decreasing? Why or why not?\r\n\r\n5. How do you find the inverse of a function algebraically?\r\n\r\n6. Show that the function [latex]f\\left(x\\right)=a-x[\/latex] is its own inverse for all real numbers [latex]a[\/latex].\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\n8.\u00a0[latex]f\\left(x\\right)=x+5[\/latex]\r\n\r\n9. [latex]f\\left(x\\right)=2-x[\/latex]\r\n\r\n10.\u00a0[latex]f\\left(x\\right)=3-x[\/latex]\r\n\r\n11.\u00a0[latex]f\\left(x\\right)=\\frac{x}{x+2}[\/latex]\r\n\r\n12.\u00a0[latex]f\\left(x\\right)=\\frac{2x+3}{5x+4}[\/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\n14.\u00a0[latex]f\\left(x\\right)={\\left(x - 6\\right)}^{2}[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)={x}^{2}-5[\/latex]\r\n\r\n16.\u00a0Given [latex]f\\left(x\\right)=\\frac{x}{2}+x[\/latex] and [latex]g\\left(x\\right)=\\frac{2x}{1-x}[\/latex]\r\n\r\na. Find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex]\r\n\r\nb. What does the answer tell us about the relationship between [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)?[\/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\n18.\u00a0[latex]f\\left(x\\right)=-3x+5[\/latex] and [latex]g\\left(x\\right)=\\frac{x - 5}{-3}[\/latex]\r\n\r\nFor the following exercises, use a graphing utility to determine whether each function is one-to-one.\r\n\r\n19. [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]\r\n\r\n20.\u00a0[latex]f\\left(x\\right)=\\sqrt[3]{3x+1}[\/latex]\r\n\r\n21. [latex]f\\left(x\\right)=-5x+1[\/latex]\r\n\r\n22.\u00a0[latex]f\\left(x\\right)={x}^{3}-27[\/latex]\r\n\r\nFor the following exercises, determine whether the graph represents a one-to-one function.\r\n\r\n23.\r\n\"Graph\r\n\r\n24.\r\n\"Graph\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\n26. Solve [latex]f\\left(x\\right)=0[\/latex].\r\n\r\n27. Find [latex]{f}^{-1}\\left(0\\right)[\/latex].\r\n\r\n28.\u00a0Solve [latex]{f}^{-1}\\left(x\\right)=0[\/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\n30. Find [latex]f\\left(6\\right)\\text{ and }{f}^{-1}\\left(2\\right)[\/latex].\r\n\r\n31. If the complete graph of [latex]f[\/latex] is shown, find the domain of [latex]f[\/latex].\r\n\r\n32. If the complete graph of [latex]f[\/latex] is shown, find the range of [latex]f[\/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\n34.\u00a0If [latex]f\\left(3\\right)=2[\/latex], find [latex]{f}^{-1}\\left(2\\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\n36.\u00a0If [latex]{f}^{-1}\\left(-2\\right)=-1[\/latex], find [latex]f\\left(-1\\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\n38.\u00a0Solve [latex]f\\left(x\\right)=3[\/latex].\r\n\r\n39. Find [latex]{f}^{-1}\\left(0\\right)[\/latex].\r\n\r\n40.\u00a0Solve [latex]{f}^{-1}\\left(x\\right)=7[\/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\nFor the following exercises, find the inverse function. Then, graph the function and its inverse.\r\n\r\n42. [latex]f\\left(x\\right)=\\frac{3}{x - 2}[\/latex]\r\n\r\n43. [latex]f\\left(x\\right)={x}^{3}-1[\/latex]\r\n\r\n44. Find the inverse function of [latex]f\\left(x\\right)=\\frac{1}{x - 1}[\/latex]. Use a graphing utility to find its domain and range. Write the domain and range in interval notation.\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\n46.\u00a0The circumference [latex]C[\/latex] of a circle is a function of its radius given by [latex]C\\left(r\\right)=2\\pi r[\/latex]. Express the radius of a circle as a function of its circumference. Call this function [latex]r\\left(C\\right)[\/latex]. Find [latex]r\\left(36\\pi \\right)[\/latex] and interpret its meaning.\r\n\r\n47. A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, [latex]t[\/latex], in hours given by [latex]d\\left(t\\right)=50t[\/latex]. Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function [latex]t\\left(d\\right)[\/latex]. Find [latex]t\\left(180\\right)[\/latex] and interpret its meaning.","rendered":"

1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?<\/p>\n

2. Why do we restrict the domain of the function [latex]f\\left(x\\right)={x}^{2}[\/latex] to find the function\u2019s inverse?<\/p>\n

3. Can a function be its own inverse? Explain.<\/p>\n

4. Are one-to-one functions either always increasing or always decreasing? Why or why not?<\/p>\n

5. How do you find the inverse of a function algebraically?<\/p>\n

6. Show that the function [latex]f\\left(x\\right)=a-x[\/latex] is its own inverse for all real numbers [latex]a[\/latex].<\/p>\n

For the following exercises, find [latex]{f}^{-1}\\left(x\\right)[\/latex] for each function.<\/p>\n

7. [latex]f\\left(x\\right)=x+3[\/latex]<\/p>\n

8.\u00a0[latex]f\\left(x\\right)=x+5[\/latex]<\/p>\n

9. [latex]f\\left(x\\right)=2-x[\/latex]<\/p>\n

10.\u00a0[latex]f\\left(x\\right)=3-x[\/latex]<\/p>\n

11.\u00a0[latex]f\\left(x\\right)=\\frac{x}{x+2}[\/latex]<\/p>\n

12.\u00a0[latex]f\\left(x\\right)=\\frac{2x+3}{5x+4}[\/latex]<\/p>\n

For 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.<\/p>\n

13. [latex]f\\left(x\\right)={\\left(x+7\\right)}^{2}[\/latex]<\/p>\n

14.\u00a0[latex]f\\left(x\\right)={\\left(x - 6\\right)}^{2}[\/latex]<\/p>\n

15. [latex]f\\left(x\\right)={x}^{2}-5[\/latex]<\/p>\n

16.\u00a0Given [latex]f\\left(x\\right)=\\frac{x}{2}+x[\/latex] and [latex]g\\left(x\\right)=\\frac{2x}{1-x}[\/latex]<\/p>\n

a. Find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex]<\/p>\n

b. What does the answer tell us about the relationship between [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)?[\/latex]<\/p>\n

For 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.<\/p>\n

17. [latex]f\\left(x\\right)=\\sqrt[3]{x - 1}[\/latex] and [latex]g\\left(x\\right)={x}^{3}+1[\/latex]<\/p>\n

18.\u00a0[latex]f\\left(x\\right)=-3x+5[\/latex] and [latex]g\\left(x\\right)=\\frac{x - 5}{-3}[\/latex]<\/p>\n

For the following exercises, use a graphing utility to determine whether each function is one-to-one.<\/p>\n

19. [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n

20.\u00a0[latex]f\\left(x\\right)=\\sqrt[3]{3x+1}[\/latex]<\/p>\n

21. [latex]f\\left(x\\right)=-5x+1[\/latex]<\/p>\n

22.\u00a0[latex]f\\left(x\\right)={x}^{3}-27[\/latex]<\/p>\n

For the following exercises, determine whether the graph represents a one-to-one function.<\/p>\n

23.
\n\"Graph<\/p>\n

24.
\n\"Graph<\/p>\n

For the following exercises, use the graph of [latex]f[\/latex] shown below.
\n\"Graph
\n25. Find [latex]f\\left(0\\right)[\/latex].<\/p>\n

26. Solve [latex]f\\left(x\\right)=0[\/latex].<\/p>\n

27. Find [latex]{f}^{-1}\\left(0\\right)[\/latex].<\/p>\n

28.\u00a0Solve [latex]{f}^{-1}\\left(x\\right)=0[\/latex].<\/p>\n

For the following exercises, use the graph of the one-to-one function shown below.
\n\"Graph
\n29. Sketch the graph of [latex]{f}^{-1}[\/latex].<\/p>\n

30. Find [latex]f\\left(6\\right)\\text{ and }{f}^{-1}\\left(2\\right)[\/latex].<\/p>\n

31. If the complete graph of [latex]f[\/latex] is shown, find the domain of [latex]f[\/latex].<\/p>\n

32. If the complete graph of [latex]f[\/latex] is shown, find the range of [latex]f[\/latex].<\/p>\n

For the following exercises, evaluate or solve, assuming that the function [latex]f[\/latex] is one-to-one.<\/p>\n

33. If [latex]f\\left(6\\right)=7[\/latex], find [latex]{f}^{-1}\\left(7\\right)[\/latex].<\/p>\n

34.\u00a0If [latex]f\\left(3\\right)=2[\/latex], find [latex]{f}^{-1}\\left(2\\right)[\/latex].<\/p>\n

35. If [latex]{f}^{-1}\\left(-4\\right)=-8[\/latex], find [latex]f\\left(-8\\right)[\/latex].<\/p>\n

36.\u00a0If [latex]{f}^{-1}\\left(-2\\right)=-1[\/latex], find [latex]f\\left(-1\\right)[\/latex].<\/p>\n

For the following exercises, use the values listed in the table below\u00a0to evaluate or solve.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
[latex]x[\/latex] <\/strong><\/td>\n [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<\/tr>\n
0<\/td>\n8<\/td>\n<\/tr>\n
1<\/td>\n0<\/td>\n<\/tr>\n
2<\/td>\n7<\/td>\n<\/tr>\n
3<\/td>\n4<\/td>\n<\/tr>\n
4<\/td>\n2<\/td>\n<\/tr>\n
5<\/td>\n6<\/td>\n<\/tr>\n
6<\/td>\n5<\/td>\n<\/tr>\n
7<\/td>\n3<\/td>\n<\/tr>\n
8<\/td>\n9<\/td>\n<\/tr>\n
9<\/td>\n1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

37. Find [latex]f\\left(1\\right)[\/latex].<\/p>\n

38.\u00a0Solve [latex]f\\left(x\\right)=3[\/latex].<\/p>\n

39. Find [latex]{f}^{-1}\\left(0\\right)[\/latex].<\/p>\n

40.\u00a0Solve [latex]{f}^{-1}\\left(x\\right)=7[\/latex].<\/p>\n

41. Use the tabular representation of [latex]f[\/latex] to create a table for [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/p>\n\n\n\n\n
[latex]x[\/latex] <\/strong><\/td>\n3<\/td>\n6<\/td>\n9<\/td>\n13<\/td>\n14<\/td>\n<\/tr>\n
[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n1<\/td>\n4<\/td>\n7<\/td>\n12<\/td>\n16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For the following exercises, find the inverse function. Then, graph the function and its inverse.<\/p>\n

42. [latex]f\\left(x\\right)=\\frac{3}{x - 2}[\/latex]<\/p>\n

43. [latex]f\\left(x\\right)={x}^{3}-1[\/latex]<\/p>\n

44. Find the inverse function of [latex]f\\left(x\\right)=\\frac{1}{x - 1}[\/latex]. Use a graphing utility to find its domain and range. Write the domain and range in interval notation.<\/p>\n

45.\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.<\/p>\n

46.\u00a0The circumference [latex]C[\/latex] of a circle is a function of its radius given by [latex]C\\left(r\\right)=2\\pi r[\/latex]. Express the radius of a circle as a function of its circumference. Call this function [latex]r\\left(C\\right)[\/latex]. Find [latex]r\\left(36\\pi \\right)[\/latex] and interpret its meaning.<\/p>\n

47. A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, [latex]t[\/latex], in hours given by [latex]d\\left(t\\right)=50t[\/latex]. Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function [latex]t\\left(d\\right)[\/latex]. Find [latex]t\\left(180\\right)[\/latex] and interpret its meaning.<\/p>\n\n\t\t\t

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