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f ( x ) = | x 3 |

Graph of f(x).
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For the following exercises, solve the absolute value equation.

| x + 4 | = 18

x = 22 ,   x = 14

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| 1 3 x + 5 | = | 3 4 x 2 |

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For the following exercises, solve the inequality and express the solution using interval notation.

| 3 x 2 | < 7

( 5 3 , 3 )

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Inverse Functions

For the following exercises, find   f 1 ( x )   for each function.

f ( x ) = x x + 2

f 1 ( x ) = 2 x x 1

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For the following exercise, find a domain on which the 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.

Given f ( x ) = x 3 5 and g ( x ) = x + 5 3 :

  1. Find   f ( g ( x ) ) and g ( f ( x ) ) .
  2. What does the answer tell us about the relationship between f ( x ) and g ( x ) ?
  1.   f ( g ( x ) ) = x and g ( f ( x ) ) = x .
  2. This tells us that f and g are inverse functions
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For the following exercises, use a graphing utility to determine whether each function is one-to-one.

f ( x ) = 1 x

The function is one-to-one.

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f ( x ) = 3 x 2 + x

The function is not one-to-one.

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If f ( 5 ) = 2 , find f 1 ( 2 ) .

5

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

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Practice test

For the following exercises, determine whether each of the following relations is a function.

y = 2 x + 8

The relation is a function.

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{ ( 2 , 1 ) , ( 3 , 2 ) , ( 1 , 1 ) , ( 0 , 2 ) }

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For the following exercises, evaluate the function f ( x ) = 3 x 2 + 2 x at the given input.

Show that the function f ( x ) = 2 ( x 1 ) 2 + 3 is not one-to-one.

The graph is a parabola and the graph fails the horizontal line test.

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Write the domain of the function f ( x ) = 3 x in interval notation.

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Given f ( x ) = 2 x 2 5 x , find f ( a + 1 ) f ( 1 ) .

2 a 2 a

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Graph the function f ( x ) = { x + 1    if 2 < x < 3     x     if   x 3

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Find the average rate of change of the function f ( x ) = 3 2 x 2 + x by finding f ( b ) f ( a ) b a .

2 ( a + b ) + 1

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For the following exercises, use the functions f ( x ) = 3 2 x 2 + x  and  g ( x ) = x to find the composite functions.

Express H ( x ) = 5 x 2 3 x 3 as a composition of two functions, f and g , where ( f g ) ( x ) = H ( x ) .

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For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.

For the following exercises, determine whether the functions are even, odd, or neither.

f ( x ) = 5 x 2 + 9 x 6

even

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f ( x ) = 5 x 3 + 9 x 5

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Graph the absolute value function f ( x ) = 2 | x 1 | + 3.

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Solve | 2 x 3 | = 17.

x = 7 and x = 10

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Solve | 1 3 x 3 | 17. Express the solution in interval notation.

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For the following exercises, find the inverse of the function.

f ( x ) = 3 x 5

f 1 ( x ) = x + 5 3

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

Graph of a cubic function.

On what intervals is the function increasing?

( , 1.1 )  and  ( 1.1 , )

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On what intervals is the function decreasing?

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Approximate the local minimum of the function. Express the answer as an ordered pair.

( 1.1 , 0.9 )

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Approximate the local maximum of the function. Express the answer as an ordered pair.

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

Graph of absolute function and step function.

Find f ( 2 ) .

f ( 2 ) = 2

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Write an equation for the piecewise function.

f ( x ) = { | x | if x 2 3 if x > 2

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

x F ( x )
0 1
1 3
2 5
3 7
4 9
5 11
6 13
7 15
8 17

Solve the equation F ( x ) = 5.

x = 2

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Is the graph increasing or decreasing on its domain?

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Is the function represented by the graph one-to-one?

yes

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Find F 1 ( 15 ) .

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Given f ( x ) = 2 x + 11 , find f 1 ( x ) .

f 1 ( x ) = x 11 2

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Practice Key Terms 1

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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