Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 1 Chapter 5 Logarithmic...

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 1

Chapter 5Logarithmic Functions


5.1 Composite Functions


Composite Function

Definition

If f and g are functions, x is in the domain of g, and g(x) is in the domain of f, then we can form the composite function f ◦ g:

(f ◦ g)(x) = f(g(x))

We say f ◦ g is the composition of f and g.


Example: Using Tables to Evaluate a Composite Function

All of the input-output pairs of functions g and f are shown in the tables below.

1. Find (f ◦ g)(0).2. Use a table to describe five input-output pairs of f ◦ g.


Solution

1. ( )( ) ( )0 ( )0f g f g)8(f

21

2. We repeat the process used in Problem 1 for the inputs 1, 2, 3, and 4 and how the work in the table.


Example: Evaluating a Composite Function

Let f(x) = 2x – 5 and g(x) = 3x + 6.

1. Find (f ◦ g)(4). 2. Find (g ◦ f)(4).


Solution

1. 2.( )( ) ( )4 ( )4f g f g

)8(1f

2( 518)

31

( )( ) ( )4 ( )4g f g f

)3(g

3( 63) 15

Let f(x) = 2x – 5 and g(x) = 3x + 6.


Composite Functions

In general, the outputs (f ◦ g)(a) and (g ◦ f)(a) may or may not be equal.


Example: Finding Equations of Composition Functions

Let f(x) = 2x and g(x) = x + 4.

1. Find an equation of f ◦ g.

2. Find an equation of g ◦ f.


Solution

1. Since g(x) = x + 4, we can substitute x + 4 for g(x) in the second step:

( )( ) ( ( ))f g x f g x

4( )f x 42x


Solution

1. We verify our work by creating a graphing calculator table for y = f(g(x)) and y = 2x + 4.


Solution

2. Since f(x) = 2x, we can substitute 2x for f(x) in the second step.

( )( ) ( ( ))g f x g f x

)2( xg

42x


Example: Evaluating and Finding an Equation of a Composite Function

Let f(x) = –3x + 8 and g(x) = 4x – 5.

1. Find (f ◦ g)(2).2. Find an equation of f ◦ g.3. Use the equation of f ◦ g to find (f ◦ g)(2).


Solution

1. ( )(2) ( (2))f g f g

)3(f

3( ) 83

1

Let f(x) = –3x + 8 and g(x) = 4x – 5.


Solution

2. Since g(x) = 4x – 5, we can substitute 4x – 5 for g(x) in the second step.

( )( ) ( ( ))f g x f g x5( )4f x 53( 84 )x

12 15 8x 12 23x

Let f(x) = –3x + 8 and g(x) = 4x – 5.


Solution

3.

So, (f ◦ g)(2) is equal to –1, which is the same result we found in Problem 1.

( )( ) ( 32 ) 2212f g 1

Let f(x) = –3x + 8 and g(x) = 4x – 5.


Example: Expressing a Function as a Composition of Two Functions

If h(x) = (7x + 3)5, find equations of the functions f and g such that h(x) = (f ◦ g)(x).


Solution

To form f(g(x)), we substitute g(x) for x in f(x). Similarly, to form (7x + 3)5, we can substitute 7x + 3 for x in x5. This suggests that g(x) = 7x + 3 and f(x) = x5. We check by performing the composition:

(f ◦ g)(x) = f(g(x)) = f(7x + 3) = (7x + 3)5


Solution

There are other possible answers. For example, g(x) = 7x and f(x) = (x + 3)5 also work:

(f ◦ g)(x) = f(g(x)) = f(7x) = (7x + 3)5


Example: Using Graphs to Evaluate a Composite Function

Refer to the graph to find (f ◦ g)(5).


Solution

The blue arrows show that g(5) = –3. So, we can substitute –3 for g(5) in the second step:

( )(5) ( (5))f g f g

3( )f

2


Example: Using a Composite Function to Model a Situation

Let f(Q) be the number of cups in Q quarts, and let g(x) be the number of ounces in x cups.

1. Find an equation of f.2. Find an equation of g.3. Find an equation of g ◦ f.4. Find (g ◦ f)(5). What does it mean in this situation?


Solution

1. Since there are 4 cups in one quart, f(Q) = 4Q.

2. Since there are 8 ounces in one cup, g(x) = 8x.

3. (g ◦ f)(Q) = (g(f(Q)) = g(4Q) = 8(4Q) = 32Q


Solution4. (g ◦ f)(5) = 32(5) = 160

The function f converts units of quarts to units of cups, and the function g converts units of cups to units of ounces, so g ◦ f converts units of quarts to units of ounces.

So, (g ◦ f)(5) = 160 means there are 160 ounces in 5 quarts.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 1 Chapter 5 Logarithmic...

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