2.1 Section Exercises

2.1 Section Exercises


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[latex]\text{ }f\left(x\right)={x}^{2}\text{ }[/latex]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?


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

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

7. [latex]f\left(x\right)=x+3[/latex]

8. [latex]f\left(x\right)=x+5[/latex]

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

10. [latex]f\left(x\right)=3-x[/latex]

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

12. [latex]f\left(x\right)=\frac{2x+3}{5x+4}[/latex]

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

13. [latex]f\left(x\right)={\left(x+7\right)}^{2}[/latex]

14. [latex]f\left(x\right)={\left(x-6\right)}^{2}[/latex]

15. [latex]f\left(x\right)={x}^{2}-5[/latex]

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

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

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

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

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


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

19. [latex]f\left(x\right)=\sqrt{x}[/latex]

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

21. [latex]f\left(x\right)=-5x+1[/latex]

22. [latex]f\left(x\right)={x}^{3}-27[/latex]

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

23. Graph of a parabola.

24. Graph of a step-function.

For the following exercises, use the graph of[latex]\text{ }f\text{ }[/latex]shown in (Figure).

Graph of a line.

Figure 11.

25.Find[latex]\text{ }f\left(0\right).[/latex]

26. Solve[latex]\text{ }f\left(x\right)=0.[/latex]

27. Find[latex]\text{ }{f}^{-1}\left(0\right).[/latex]

28. Solve[latex]\text{ }{f}^{-1}\left(x\right)=0.[/latex]

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

Graph of a square root function.

Figure 12.

29. Sketch the graph of[latex]\text{ }{f}^{-1}.\text{ }[/latex]

30. Find[latex]\text{ }f\left(6\right)\text{ and }{f}^{-1}\left(2\right).[/latex]

31. If the complete graph of[latex]\text{ }f\text{ }[/latex]is shown, find the domain of[latex]\text{ }f.\text{ }[/latex]

32. If the complete graph of[latex]\text{ }f\text{ }[/latex]is shown, find the range of[latex]\text{ }f.[/latex]


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

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

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

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

36. If[latex]\text{ }{f}^{-1}\left(-2\right)=-1,\text{ }[/latex]find[latex]\text{ }f\left(-1\right).[/latex]

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

Table 6

[latex]x[/latex] [latex]f\left(x\right)[/latex]
0 8
1 0
2 7
3 4
4 2
5 6
6 5
7 3
8 9
9 1

37. Find[latex]\text{ }f\left(1\right).[/latex]

38. Solve[latex]\text{ }f\left(x\right)=3.[/latex]

39. Find[latex]\text{ }{f}^{-1}\left(0\right).[/latex]

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

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

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


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

42. [latex]f\left(x\right)=\frac{3}{x-2}[/latex]

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

44. Find the inverse function of[latex]\text{ }f\left(x\right)=\frac{1}{x-1}.\text{ }[/latex]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[latex]\text{ }x\text{ }[/latex]degrees Celsius to[latex]\text{ }y\text{ }[/latex]degrees Fahrenheit, we use the formula[latex]\text{ }f\left(x\right)=\frac{9}{5}x+32.\text{ }[/latex]Find the inverse function, if it exists, and explain its meaning.

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