Problem Set 7: Inverse Functions

1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

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

3. Can a function be its own inverse? Explain.

4. Are one-to-one functions either always increasing or always decreasing? Why or why not?

5. How do you find the inverse of a function algebraically?

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

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

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

8. [latex]f\left(x\right)=x+5[/latex]

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

10. [latex]f\left(x\right)=3-x[/latex]

11. [latex]f\left(x\right)=\frac{x}{x+2}[/latex]

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

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.

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

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

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

16. Given [latex]f\left(x\right)=\frac{x}{2}+x[/latex] and [latex]g\left(x\right)=\frac{2x}{1-x}[/latex]

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

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

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.

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

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

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

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

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

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

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

For the following exercises, determine whether the graph represents a one-to-one function.

Graph of an upright parabola with vertex at (10, -10), passing through (0,0) and (25,0)

Graph of a step-function, with y = -5 for {x|-10<=x,0} and y = 0 for {x|0<=x<10}

For the following exercises, use the graph of [latex]f[/latex] shown below.
Graph of the line y = (-3/2)x + 3
25. Find [latex]f\left(0\right)[/latex].

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

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

28. Solve [latex]{f}^{-1}\left(x\right)=0[/latex].

For the following exercises, use the graph of the one-to-one function shown below.
Graph of a square root function for {x|x>=2}
29. Sketch the graph of [latex]{f}^{-1}[/latex].

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

31. If the complete graph of [latex]f[/latex] is shown, find the domain of [latex]f[/latex].

32. If the complete graph of [latex]f[/latex] is shown, find the range of [latex]f[/latex].

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

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

34. If [latex]f\left(3\right)=2[/latex], find [latex]{f}^{-1}\left(2\right)[/latex].

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

36. If [latex]{f}^{-1}\left(-2\right)=-1[/latex], find [latex]f\left(-1\right)[/latex].

For the following exercises, use the values listed in the table below to evaluate or solve.

[latex]x[/latex] [latex]f\left(x\right)[/latex]
0 8
1 0
2 7
3 4
4 2
5 6
6 5
7 3
8 9
9 1

37. Find [latex]f\left(1\right)[/latex].

38. Solve [latex]f\left(x\right)=3[/latex].

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

40. Solve [latex]{f}^{-1}\left(x\right)=7[/latex].

41. Use the tabular representation of [latex]f[/latex] to create a table for [latex]{f}^{-1}\left(x\right)[/latex].

[latex]x[/latex] 3 6 9 13 14
[latex]f\left(x\right)[/latex] 1 4 7 12 16

For the following exercises, find the inverse function. Then, graph the function and its inverse.

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

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

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.

45. To 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.

46. The 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.

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.