# 1.7E: Exercises

- Page ID
- 17377

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### Section Exercises

**Verbal**

Exercise 1.7.1

Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

**Answer:**- 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 \(y\)-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 \(y\)-values repeat and the function is one-to-one.

Exercise 1.7.2

Why do we restrict the domain of the function \(f(x)=x^2\) to find the function’s inverse?

Exercise 1.7.3

Can a function be its own inverse? Explain.

**Answer:**- Yes. For example, \(f(x)=\frac{1}{x}\) is its own inverse.

Exercise 1.7.4

Are one-to-one functions either always increasing or always decreasing? Why or why not?

Exercise 1.7.5

How do you find the inverse of a function algebraically?

**Answer:**- Given a function \(y=f(x)\), solve for \(x\) in terms of \(y\). Interchange the \(x\) and \(y\). Solve the new equation for \(y\). The expression for \(y\) is the inverse, \(y=f^{-1}(x)\).

**Algebraic**

Exercise 1.7.6

Show that the function \(f(x)=a−x\) is its own inverse for all real numbers \(a\).

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

Exercise 1.7.7

\(f(x)=x+3\)

**Answer:**- \(f^{-1}(x)=x−3\)

Exercise 1.7.8

\(f(x)=x+5\)

Exercise 1.7.9

\(f(x)=2−x\)

**Answer:**- \(f^{-1}(x)=2−x\)

Exercise 1.7.10:

\(f(x)=3−x\)

Exercise 1.7.11

\(f(x)=\frac{x}{x+2}\)

**Answer:**- \(f^{-1}(x)=\frac{−2x}{x−1}\)

Exercise 1.7.12

\(f(x)=\frac{2x+3}{5x+4}\)

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

Exercise 1.7.13

\(f(x)=(x+7)^2\)

**Answer:**- domain of \(f(x)\): \(\left[−7,\infty\right)\); \(f^{-1}(x)=\sqrt{x}−7\)

Exercise 1.7.14

\(f(x)=(x−6)^2\)

Exercise 1.7.15

\(f(x)=x^2−5\)

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

Exercise 1.7.16

Given \(f(x)=\frac{x}{2+x}\) and \(g(x)=\frac{2x}{1-x}:\)

a. Find \(f(g(x))\) and \(g(f(x))\).

b. What does the answer tell us about the relationship between \(f(x)\) and \(g(x)?\)

**Answer:**- a. \(f(g(x))=x\) and \(g(f(x))=x\).

b. This tells us that \(f\) and \(g\) are inverse functions

For the following exercises, use function composition to verify that \(f(x)\) and \(g(x)\) are inverse functions.

Exercise 1.7.17

\(f(x)=\sqrt[3]{x-1}\) and \(g(x)=x^3+1\)

**Answer:**- \(f(g(x))=x\), \(g(f(x))=x\)

Exercise 1.7.18

\(f(x)=−3x+5\) and \(g(x)=\frac{x-5}{-3}\)

**Graphical**

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

Exercise 1.7.19

\(f(x)=\sqrt{x}\)

**Answer:**- one-to-one

Exercise 1.7.20

\(f(x)=\sqrt[3]{3x+1}\)

Exercise 1.7.21

\(f(x)=−5x+1\)

**Answer:**- one-to-one

Exercise 1.7.22

\(f(x)=x^3−27\)

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

Exercise 1.7.23

**Answer:**- not one-to-one

Exercise 1.7.24

For the following exercises, use the graph of \(f\) shown in **Figure 1.7.11**.

**Figure 1.7.11:** Graph of a line

Exercise 1.7.25

Find \(f(0)\).

**Answer:**- 3

Exercise 1.7.26

Solve \(f(x)=0\).

Exercise 1.7.27

Find \(f^{-1}(0)\).

**Answer:**- 2

Exercise 1.7.28

Solve \(f^{-1}(x)=0\).

For the following exercises, use the graph of the one-to-one function shown in **Figure 1.7.12**.

**Figure 1.7.12: **Graph of a square root function.

Exercise 1.7.29

Sketch the graph of \(f^{-1}\).

**Answer:**

Exercise 1.7.30

Find \(f(6)\) and \(f^{-1}(2)\).

Exercise 1.7.31

If the complete graph of \(f\) is shown, find the domain of \(f\).

**Answer:**- \([2,10]\)

Exercise 1.7.32

If the complete graph of \(f\) is shown, find the range of \(f\)

**Numeric**

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one.

Exercise 1.7.33

If \(f(6)=7\), find \(f^{-1}(7)\).

**Answer:**- 6

Exercise 1.7.34

If \(f(3)=2\), find \(f^{-1}(2)\).

Exercise 1.7.35

If \(f^{-1}(−4)=−8\), find \(f(−8)\).

**Answer:**- -4

Exercise 1.7.36

If \(f^{-1}(−2)\)=−1, find \(f(−1)\).

For the following exercises, use the values listed in **Table 1.7.6** to evaluate or solve.

\(x\) | \(f(x)\) |

0 | 8 |

1 | 0 |

2 | 7 |

3 | 4 |

4 | 2 |

5 | 6 |

6 | 5 |

7 | 3 |

8 | 9 |

9 | 1 |

**Table 1.7.6**

Exercise 1.7.37

Find \(f(1)\).

**Answer:**- 0

Exercise 1.7.38

Solve \(f(x)=3\).

Exercise 1.7.39

Find \(f^{-1}(0)\).

**Answer:**- 1

Exercise 1.7.40

Solve \(f^{-1}(x)=7\).

Exercise 1.7.41

Use the tabular representation of \(f\) in **Table 1.7.7** to create a table for f^{-1}(x).

\(x\) |
3 | 6 | 9 | 13 | 14 |
---|---|---|---|---|---|

\(f(x)\) |
1 | 4 | 7 | 12 | 16 |

**Table 1.7.7**

**Answer:**-
\(x\)

1 4 7 12 16 \(f^{-1}(x)\)

3 6 9 13 14

**Technology**

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

Exercise 1.7.42

\(f(x)=\dfrac{3}{x-2}\)

Exercise 1.7.43

\(f(x)=x^3−1\)

**Answer:**-
\(f^{-1}(x)=(1+x)^{1/3}\)

Exercise 1.7.44

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

**Real-World Applications**

Exercise 1.7.45

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

**Answer:**- \(f^{-1}(x)=\frac{5}{9}(x−32)\). Given the Fahrenheit temperature, \(x\), this formula allows you to calculate the Celsius temperature.

Exercise 1.7.46

The circumference \(C\) of a circle is a function of its radius given by \(C(r)=2{\pi}r\). Express the radius of a circle as a function of its circumference. Call this function \(r(C)\). Find \(r(36\pi)\) and interpret its meaning.

Exercise 1.7.47

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, \(t\), in hours given by \(d(t)=50t\). Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function \(t(d)\). Find \(t(180)\) and interpret its meaning.

**Answer:**- \(t(d)=\frac{d}{50}\), \(t(180)=\frac{180}{50}\). The time for the car to travel 180 miles is 3.6 hours.