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1.6 Inverse functions  (Page 7/10)

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

f ( x ) = x − 1 3   and   g ( x ) = x 3 + 1

  f ( g ( x ) ) = x ,   g ( f ( x ) ) = x

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

Graphical

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

f ( x ) = x

one-to-one

f ( x ) = 3 x + 1 3

f ( x ) = −5 x + 1

one-to-one

f ( x ) = x 3 − 27

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

Graph of a parabola.

not one-to-one

Graph of a step-function.

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

Graph of a line.

Find   f ( 0 ) .

3

Solve   f ( x ) = 0.

Find   f − 1 ( 0 ) .

2

Solve   f − 1 ( x ) = 0.

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   f − 1 .  

Graph of a square root function and its inverse.

Find   f ( 6 )  and  f − 1 ( 2 ) .

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

[ 2 , 10 ]

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.

If   f ( 6 ) = 7 ,   find     f − 1 ( 7 ) .

6

If   f ( 3 ) = 2 ,   find   f − 1 ( 2 ) .

If   f − 1 ( − 4 ) = − 8 ,   find   f ( − 8 ) .

− 4

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

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

Solve   f ( x ) = 3.

Find   f − 1 ( 0 ) .

  1  

Solve   f − 1 ( x ) = 7.

Use the tabular representation of   f   in [link] to create a table for   f − 1 ( x ) .

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

f ( x ) = 3 x − 2

f ( x ) = x 3 − 1

f − 1 ( x ) = ( 1 + x ) 1 / 3

Graph of a cubic function and its inverse.

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

Real-world applications

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

f − 1 ( x ) = 5 9 ( x − 32 ) .   Given the Fahrenheit temperature,   x ,   this formula allows you to calculate the Celsius temperature.

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

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 ) = 50 t .   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 ) = d 50 ,   t ( 180 ) = 180 50 .   The time for the car to travel 180 miles is 3.6 hours.

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

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

y 2 + 4 = x ,   for   x   the independent variable and   y   the dependent variable

not a function

Is the graph in [link] a function?

Graph of a parabola.

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

f ( x ) = − 2 x 2 + 3 x

f ( − 3 ) = − 27 ; f ( 2 ) = − 2 ; f ( − a ) = − 2 a 2 − 3 a ;
− f ( a ) = 2 a 2 − 3 a ; f ( a + h ) = − 2 a 2 + 3 a − 4 a h + 3 h − 2 h 2

f ( x ) = 2 | 3 x − 1 |

For the following exercises, determine whether the functions are one-to-one.
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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