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]
9.
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]
29.
31. [latex]\left[2,10\right][/latex]
33. [latex]6[/latex]
35. [latex]-4[/latex]
37. [latex]0[/latex]
39. [latex]1[/latex]
41.
[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]
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.