## 2.1 Section Exercises

### 2.1 Section Exercises

#### Verbal

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$\text{ }f\left(x\right)={x}^{2}\text{ }$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?

#### Algebraic

6. Show that the function$\text{ }f\left(x\right)=a-x\text{ }$is its own inverse for all real numbers$\text{ }a.\text{ }$

For the following exercises, find$\text{ }{f}^{-1}\left(x\right)\text{ }$for each function.

7. $f\left(x\right)=x+3$

8. $f\left(x\right)=x+5$

9. $f\left(x\right)=2-x$

10. $f\left(x\right)=3-x$

11. $f\left(x\right)=\frac{x}{x+2}$

12. $f\left(x\right)=\frac{2x+3}{5x+4}$

For the following exercises, find a domain on which each function$\text{ }f\text{ }$is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of$\text{ }f\text{ }$restricted to that domain.

13. $f\left(x\right)={\left(x+7\right)}^{2}$

14. $f\left(x\right)={\left(x-6\right)}^{2}$

15. $f\left(x\right)={x}^{2}-5$

16. Given$\text{ }f\left(x\right)=\frac{x}{2+x}\text{ }$and$\text{ }g\left(x\right)=\frac{2x}{1-x}:$

1. Find$\text{ }f\left(g\left(x\right)\right)\text{ }$and$\text{ }g\left(f\left(x\right)\right).$
2. What does the answer tell us about the relationship between$\text{ }f\left(x\right)\text{ }$and$\text{ }g\left(x\right)?$

For the following exercises, use function composition to verify that$\text{ }f\left(x\right)\text{ }$and$\text{ }g\left(x\right)\text{ }$are inverse functions.

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

18. $f\left(x\right)=-3x+5\text{ }$and$\text{ }g\left(x\right)=\frac{x-5}{-3}$

#### Graphical

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

19. $f\left(x\right)=\sqrt{x}$

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

21. $f\left(x\right)=-5x+1$

22. $f\left(x\right)={x}^{3}-27$

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

23.

24.

For the following exercises, use the graph of$\text{ }f\text{ }$shown in (Figure).

Figure 11.

25.Find$\text{ }f\left(0\right).$

26. Solve$\text{ }f\left(x\right)=0.$

27. Find$\text{ }{f}^{-1}\left(0\right).$

28. Solve$\text{ }{f}^{-1}\left(x\right)=0.$

For the following exercises, use the graph of the one-to-one function shown in (Figure).

Figure 12.

29. Sketch the graph of$\text{ }{f}^{-1}.\text{ }$

30. Find$\text{ }f\left(6\right)\text{ and }{f}^{-1}\left(2\right).$

31. If the complete graph of$\text{ }f\text{ }$is shown, find the domain of$\text{ }f.\text{ }$

32. If the complete graph of$\text{ }f\text{ }$is shown, find the range of$\text{ }f.$

#### Numeric

For the following exercises, evaluate or solve, assuming that the function$\text{ }f\text{ }$is one-to-one.

33. If$\text{ }f\left(6\right)=7,\text{ }$find$\text{ }\text{ }{f}^{-1}\left(7\right).$

34. If$\text{ }f\left(3\right)=2,\text{ }$find$\text{ }{f}^{-1}\left(2\right).$

35. If$\text{ }{f}^{-1}\left(-4\right)=-8,\text{ }$find$\text{ }f\left(-8\right).$

36. If$\text{ }{f}^{-1}\left(-2\right)=-1,\text{ }$find$\text{ }f\left(-1\right).$

For the following exercises, use the values listed in (Figure) to evaluate or solve.

 Table 6 $x$ $f\left(x\right)$ 0 8 1 0 2 7 3 4 4 2 5 6 6 5 7 3 8 9 9 1

37. Find$\text{ }f\left(1\right).$

38. Solve$\text{ }f\left(x\right)=3.$

39. Find$\text{ }{f}^{-1}\left(0\right).$

40. Solve$\text{ }{f}^{-1}\left(x\right)=7.$

41. Use the tabular representation of$\text{ }f\text{ }$in (Figure) to create a table for$\text{ }{f}^{-1}\left(x\right).$

 Table 7 $x$ 3 6 9 13 14 $f\left(x\right)$ 1 4 7 12 16

#### Technology

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

42. $f\left(x\right)=\frac{3}{x-2}$

43. $f\left(x\right)={x}^{3}-1$

44. Find the inverse function of$\text{ }f\left(x\right)=\frac{1}{x-1}.\text{ }$Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

#### Real-World Applications

45. To convert from$\text{ }x\text{ }$degrees Celsius to$\text{ }y\text{ }$degrees Fahrenheit, we use the formula$\text{ }f\left(x\right)=\frac{9}{5}x+32.\text{ }$Find the inverse function, if it exists, and explain its meaning.

46. The circumference$\text{ }C\text{ }$of a circle is a function of its radius given by$\text{ }C\left(r\right)=2\pi r.\text{ }$Express the radius of a circle as a function of its circumference. Call this function$\text{ }r\left(C\right).\text{ }$Find$\text{ }r\left(36\pi \right)\text{ }$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,$\text{ }t,\text{ }$in hours given by$\text{ }d\left(t\right)=50t.\text{ }$Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function$\text{ }t\left(d\right).\text{ }$Find$\text{ }t\left(180\right)\text{ }$and interpret its meaning.