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?

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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 $\text{ }y$ -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 $\text{ }y$ -values repeat and the 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.

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Yes. For example, $\text{ }f\left(x\right)=\frac{1}{x}\text{ }$ is its own inverse.

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?

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Given a function $\text{ }y=f\left(x\right),\text{ }$ solve for $\text{ }x\text{ }$ in terms of $\text{ }y.\text{ }$ The expression for $\text{ }x\text{ }$ is the inverse, $\text{ }{f}^{-1}\left(y\right).$

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

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f^{- 1} (y) = y - 3

${f}^{-1}\left(y\right)=y-3$

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

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

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${f}^{-1}\left(y\right)=2-y$

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

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

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${f}^{-1}\left(y\right)=\frac{-2y}{y-1}$

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

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domain of $f\left(x\right):\text{ }\left[-7,\infty \right);\text{ }{f}^{-1}\left(y\right)=\sqrt{y}-7$

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

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

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domain of $\text{ }f\left(x\right):\text{ }\left[0,\infty \right);\text{ }{f}^{-1}\left(y\right)=\sqrt{y+5}$

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

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

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a. $\text{ }f\left(g\left(x\right)\right)=x\text{ }$ and $\text{ }g\left(f\left(x\right)\right)=x.\text{ }$ b. This tells us that $\text{ }f\text{ }$ and $\text{ }g\text{ }$ are inverse functions

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 $\text{ }g\left(x\right)={x}^{3}+1$

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$f\left(g\left(x\right)\right)=x,\text{ }g\left(f\left(x\right)\right)=x$

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

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

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

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

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

23.

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24.

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

Figure 11.

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

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$3$

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

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

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$2$

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 $\text{ }{f}^{-1}.\text{ }$

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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 $\text{ }f.\text{ }$

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$\left[2,10\right]$

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 $\text{ }\text{ }{f}^{-1}\left(7\right).$

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$6$

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 $\text{ }f\left(-8\right).$

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$-4$

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

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$0$

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

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

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$\text{ }1\text{ }$

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

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$x$	1	4	7	12	16
${f}^{-1}\left(x\right)$	3	6	9	13	14

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

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${f}^{-1}\left(x\right)={\left(1+x\right)}^{1/3}$

Graph of a cubic function and its inverse.

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.

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${f}^{-1}\left(x\right)=\frac{5}{9}\left(x-32\right).\text{ }$ Given the Fahrenheit temperature, $\text{ }x,\text{ }$ this formula allows you to calculate the Celsius temperature.

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.

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$t\left(d\right)=\frac{d}{50},\text{ }$ $t\left(180\right)=\frac{180}{50}.\text{ }$ The time for the car to travel 180 miles is 3.6 hours.

Exponential and Logarithmic Functions