Solutions 7: Inverse Functions

Solutions to Try Its

1. [latex]h\left(2\right)=6[/latex]

2. Yes

3. Yes

4. The domain of function [latex]{f}^{-1}[/latex] is [latex]\left(-\infty \text{,}-2\right)[/latex] and the range of function [latex]{f}^{-1}[/latex] is [latex]\left(1,\infty \right)[/latex].

5. a. [latex]f\left(60\right)=50[/latex]. In 60 minutes, 50 miles are traveled.

b. [latex]{f}^{-1}\left(60\right)=70[/latex]. To travel 60 miles, it will take 70 minutes.

6. a. 3; b. 5.6

7. [latex]x=3y+5[/latex]

8. [latex]{f}^{-1}\left(x\right)={\left(2-x\right)}^{2};\text{domain}\text{of}f:\left[0,\infty \right);\text{domain}\text{of}{f}^{-1}:\left(-\infty ,2\right][/latex]

Graph of f(x) and f^(-1)(x).

Solutions to Odd-Numbered Exercises

1. Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that [latex]y[/latex] -values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no [latex]y[/latex] -values repeat and the function is one-to-one.

3. Yes. For example, [latex]f\left(x\right)=\frac{1}{x}[/latex] is its own inverse.

5. Given a function [latex]y=f\left(x\right)[/latex], solve for [latex]x[/latex] in terms of [latex]y[/latex]. Interchange the [latex]x[/latex] and [latex]y[/latex]. Solve the new equation for [latex]y[/latex]. The expression for [latex]y[/latex] is the inverse, [latex]y={f}^{-1}\left(x\right)[/latex].

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

9. [latex]{f}^{-1}\left(x\right)=2-x[/latex]

11. [latex]{f}^{-1}\left(x\right)=\frac{-2x}{x - 1}[/latex]

13. domain of [latex]f\left(x\right):\left[-7,\infty \right);{f}^{-1}\left(x\right)=\sqrt{x}-7[/latex]

15. domain of [latex]f\left(x\right):\left[0,\infty \right);{f}^{-1}\left(x\right)=\sqrt{x+5}[/latex]

17. [latex] f\left(g\left(x\right)\right)=x,g\left(f\left(x\right)\right)=x[/latex]

19. one-to-one

21. one-to-one

23. not one-to-one

25. [latex]3[/latex]

27. [latex]2[/latex]

Graph of a square root function and its inverse.

31. [latex]\left[2,10\right][/latex]

33. [latex]6[/latex]

35. [latex]-4[/latex]

37. [latex]0[/latex]

39. [latex]1[/latex]


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

43. [latex]{f}^{-1}\left(x\right)={\left(1+x\right)}^{1/3}[/latex]
Graph of a cubic function and its inverse.

45. [latex]{f}^{-1}\left(x\right)=\frac{5}{9}\left(x - 32\right)[/latex]. Given the Fahrenheit temperature, [latex]x[/latex], this formula allows you to calculate the Celsius temperature.

47. [latex]t\left(d\right)=\frac{d}{50}[/latex], [latex]t\left(180\right)=\frac{180}{50}[/latex]. The time for the car to travel 180 miles is 3.6 hours.