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1.7E: Exercises

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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)=x2 to find the function’s inverse?

Exercise 1.7.3

Can a function be its own inverse? Explain.

Answer:
Yes. For example, f(x)=1x 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=f1(x).

Algebraic

Exercise 1.7.6

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

For the following exercises, find f1(x) for each function.

Exercise 1.7.7

f(x)=x+3

Answer:
f1(x)=x3

Exercise 1.7.8

f(x)=x+5

Exercise 1.7.9

f(x)=2x

Answer:
f1(x)=2x

Exercise 1.7.10:

f(x)=3x

Exercise 1.7.11

f(x)=xx+2

Answer:
f1(x)=2xx1

Exercise 1.7.12

f(x)=2x+35x+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): [7,); f1(x)=x7

Exercise 1.7.14

f(x)=(x6)2

Exercise 1.7.15

f(x)=x25

Answer:
domain of f(x): [0,); f1(x)=x+5

Exercise 1.7.16

Given f(x)=x2+x and g(x)=2x1x:

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)=3x1 and g(x)=x3+1

Answer:
f(g(x))=x, g(f(x))=x

Exercise 1.7.18

f(x)=3x+5 and g(x)=x53

Graphical

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

Exercise 1.7.19

f(x)=x

Answer:
one-to-one

Exercise 1.7.20

f(x)=33x+1

Exercise 1.7.21

f(x)=5x+1

Answer:
one-to-one

Exercise 1.7.22

f(x)=x327

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

Exercise 1.7.23

Graph of a parabola.

Answer:
not one-to-one

Exercise 1.7.24

Graph of a step-function.

For the following exercises, use the graph of f shown in Figure 1.7.11.

Graph of a line.

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 f1(0).

Answer:
2

Exercise 1.7.28

Solve f1(x)=0.

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

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

Exercise 1.7.29

Sketch the graph of f1.

Answer:

Graph of a square root function and its inverse.

Exercise 1.7.30

Find f(6) and f1(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 f1(7).

Answer:
6

Exercise 1.7.34

If f(3)=2, find f1(2).

Exercise 1.7.35

If f1(4)=8, find f(8).

Answer:
-4

Exercise 1.7.36

If f1(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 f1(0).

Answer:
1

Exercise 1.7.40

Solve f1(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

f1(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)=3x2

Exercise 1.7.43

f(x)=x31

Answer:

f1(x)=(1+x)1/3

Graph of a cubic function and its inverse.

Exercise 1.7.44

Find the inverse function of f(x)=1x1. 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)=95x+32. Find the inverse function, if it exists, and explain its meaning.

Answer:
f1(x)=59(x32). 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)=2πr. Express the radius of a circle as a function of its circumference. Call this function r(C). Find r(36π) 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)=d50, t(180)=18050. The time for the car to travel 180 miles is 3.6 hours.

1.7E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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