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] 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 |
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.Licenses & Attributions
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- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..
