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

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

 f(g(x))=x,g(f(x))=x

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

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Graphical

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

f(x)=x

one-to-one

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f(x)=33x+1

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f(x)=−5x+1

one-to-one

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f(x)=x327

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

For the following exercises, use the graph of f shown in [link] .

Graph of a line.

Find f(0).

3

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Solve f(x)=0.

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Find f1(0).

2

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Solve f1(x)=0.

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For the following exercises, use the graph of the one-to-one function shown in [link] .

Graph of a square root function.

Sketch the graph of f1.

Graph of a square root function and its inverse.
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Find f(6) and f1(2).

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

[2,10]

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If the complete graph of f is shown, find the range of f.

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Numeric

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

If f(6)=7, find f1(7).

6

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

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

4

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If f1(2)=1, find f(1).

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For the following exercises, use the values listed in [link] 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

Find f(1).

0

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Solve f(x)=3.

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Find f1(0).

1

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Solve f1(x)=7.

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Use the tabular representation of f in [link] to create a table for f1(x).

x 3 6 9 13 14
f(x) 1 4 7 12 16
x 1 4 7 12 16
f1(x) 3 6 9 13 14
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Technology

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

f(x)=3x2

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f(x)=x31

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

Graph of a cubic function and its inverse.
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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.

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Real-world applications

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.

f1(x)=59(x32). Given the Fahrenheit temperature, x, this formula allows you to calculate the Celsius temperature.

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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.

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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.

t(d)=d50, t(180)=18050. The time for the car to travel 180 miles is 3.6 hours.

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Chapter review exercises

Functions and Function Notation

For the following exercises, determine whether the relation is a function.

{(a,b),(c,d),(e,d)}

function

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{(5,2),(6,1),(6,2),(4,8)}

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y2+4=x, for x the independent variable and y the dependent variable

not a function

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Is the graph in [link] a function?

Graph of a parabola.
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For the following exercises, evaluate the function at the indicated values: f(3);f(2);f(a);f(a);f(a+h).

f(x)=2x2+3x

f(3)=27; f(2)=2; f(a)=2a23a;
f(a)=2a23a; f(a+h)=2a2+3a4ah+3h2h2

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f(x)=2|3x1|

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For the following exercises, determine whether the functions are one-to-one.

Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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