Chapter 3: Transformations of Graphs and Data

Lesson 7: Compositions of Functions

Mrs. Parziale

Composite Functions:

• The composite of two functions g with f is written g ◦ for g( f(x)).

• The domain of g(f(x)) is the set of x-values in the domain of f for which f(x) is in the domain of g. – Simply put – the set of answers from f(x) become

the input for the function g.

Example 1:

• Let and

(a) Find f(g(3)) (b) Find g(f(-2))(start with g(3))

( ) 2 1f x x 2( ) 4g x x

Draw a mapping for g(f(x))

( ) 2 1f x x 2( ) 4g x x-2-1012345

-3-113579

11

3644

36100196324484

Example 2:

• Let and (a) Find a rule for

( ) 2 1f x x 2( ) 4g x x

f g

Example 2:

• Let and (b) Find a rule for

( ) 2 1f x x 2( ) 4g x x

g f

Find the domain of the following:

• f ________________________

• g ________________________

• ____________________

• ____________________

( ) 2 1f x x 2( ) 4g x x

f g

g f

Example 3:

• Let and Find f(g(4))

g(f(4))

2( )f x x1

( )3 1

g xx

(4)f g

Find and graph the formulas for andf g g f

2( )f x x 1( )

3 1g x

x

Since the rules do not produce the same results, we can conclude that composite functions are not commutative.

f ◦ g g ◦ f

Find and graph the formulas for andf g g f

2( )f x x1

( )3 1

g xx

Find the domain of the following:

• f ________________________

• g ________________________

• ____________________

• ____________________

2( )f x x 1( )

3 1g x

x

f g

g f

Closure

• What is a composite function?• How do you find ?• Given the following:

• Find for x = 3

g f

g f

2( ) 2 ( )f x x g x x