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1.6 Inverse functions  (Page 9/10)

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f ( x ) = | x − 5 |

f ( x ) = − | x − 3 |

Graph of f(x).

f ( x ) = | 2 x − 4 |

For the following exercises, solve the absolute value equation.

| x + 4 | = 18

x = − 22 ,   x = 14

| 1 3 x + 5 | = | 3 4 x − 2 |

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

| 3 x − 2 | < 7

( − 5 3 , 3 )

| 1 3 x − 2 | ≤ 7

Inverse Functions

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

f ( x ) = 9 + 10 x

f ( x ) = x x + 2

f − 1 ( x ) = − 2 x x − 1

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.

f ( x ) = x 2 + 1

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

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.

f ( x ) = − 3 x 2 + x

The function is not one-to-one.

If f ( 5 ) = 2 , find f − 1 ( 2 ) .

5

If f ( 1 ) = 4 , find f − 1 ( 4 ) .

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.

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

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

f ( −2 )

−16

  f ( a )  

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.

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

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

2 a 2 − a

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

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

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

( g ∘ f ) ( x )

( g ∘ f ) ( 1 )

2

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

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

f ( x ) = x + 6 − 1

Graph of f(x).

f ( x ) = 1 x + 2 − 1

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

f ( x ) = − 5 x 2 + 9 x 6

even

f ( x ) = − 5 x 3 + 9 x 5

f ( x ) = 1 x

odd

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

Solve | 2 x − 3 | = 17.

x = − 7   and   x = 10

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

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

f ( x ) = 3 x − 5

f − 1 ( x ) = x + 5 3

f ( x ) = 4 x + 7

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 , ∞ )

On what intervals is the function decreasing?

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

( 1.1 , − 0.9 )

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

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

Find   f ( −2 ) .

Write an equation for the piecewise function.

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

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

Find   F ( 6 ) .

Solve the equation   F ( x ) = 5.

x = 2

Is the graph increasing or decreasing on its domain?

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

yes

Find   F − 1 ( 15 ) .

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

f − 1 ( x ) = − x − 11 2

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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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