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Study Guides > College Algebra

Solutions

Solutions to Try Its

1. h(2)=6h\left(2\right)=6 2. Yes 3. Yes 4. The domain of function f1{f}^{-1} is (,2)\left(-\infty \text{,}-2\right) and the range of function f1{f}^{-1} is (1,)\left(1,\infty \right). 5. a. f(60)=50f\left(60\right)=50. In 60 minutes, 50 miles are traveled. b. f1(60)=70{f}^{-1}\left(60\right)=70. To travel 60 miles, it will take 70 minutes. 6. a. 3; b. 5.6 7. x=3y+5x=3y+5 8. f1(x)=(2x)2;domainoff:[0,);domainoff1:(,2]{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]\\ 9. 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 yy -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 yy -values repeat and the function is one-to-one. 3. Yes. For example, f(x)=1xf\left(x\right)=\frac{1}{x}\\ is its own inverse. 5. Given a function y=f(x)y=f\left(x\right), solve for xx in terms of yy. Interchange the xx and yy. Solve the new equation for yy. The expression for yy is the inverse, y=f1(x)y={f}^{-1}\left(x\right). 7. f1(x)=x3{f}^{-1}\left(x\right)=x - 3 9. f1(x)=2x{f}^{-1}\left(x\right)=2-x 11. f1(x)=2xx1{f}^{-1}\left(x\right)=\frac{-2x}{x - 1}\\ 13. domain of f(x):[7,);f1(x)=x7f\left(x\right):\left[-7,\infty \right);{f}^{-1}\left(x\right)=\sqrt{x}-7 15. domain of f(x):[0,);f1(x)=x+5f\left(x\right):\left[0,\infty \right);{f}^{-1}\left(x\right)=\sqrt{x+5}\\ 17. f(g(x))=x,g(f(x))=x f\left(g\left(x\right)\right)=x,g\left(f\left(x\right)\right)=x 19. one-to-one 21. one-to-one 23. not one-to-one 25. 33 27. 22 29. Graph of a square root function and its inverse. 31. [2,10]\left[2,10\right] 33. 66 35. 4-4 37. 00 39. 11 41.
xx 1 4 7 12 16
f1(x){f}^{-1}\left(x\right)\\ 3 6 9 13 14
43. f1(x)=(1+x)1/3{f}^{-1}\left(x\right)={\left(1+x\right)}^{1/3}\\ Graph of a cubic function and its inverse. 45. f1(x)=59(x32){f}^{-1}\left(x\right)=\frac{5}{9}\left(x - 32\right)\\. Given the Fahrenheit temperature, xx, this formula allows you to calculate the Celsius temperature. 47. t(d)=d50t\left(d\right)=\frac{d}{50}\\, t(180)=18050t\left(180\right)=\frac{180}{50}\\. The time for the car to travel 180 miles is 3.6 hours.

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