1. Section 6.1
Composite Functions

2. 1. Form a Composition Function

3. Composite Function
The composite function, denoted by f  g
(read as “ composed with ) or g  f
(read as “ composed with ), is defined
by
( f  g )( x)  f ( g ( x))
or
( g  f )( x)  g ( f ( x))

4. Evaluating a Composite
Function
Example 1: Evaluate the expression
using the values in the table.
X -1 0 1 2 3
2 -1 -2 -1 2
3 2 3 6 11
a. ( f  g )(-1)  f (g (-1))  f (3)  2

5. 1. Find the Composition
Function

6. For the given functions and , find:
(a) ( f  g )(-1) (b) ( g  f )(0) (c) ( g  g )(1) (d ) ( f  f )(2)
1
f ( x)  2 x  1 g ( x)  x  3
2
1 
(a) ( f  g )(-1)  f ( g (-1))  f  (-1)  3 
2 
 7  7
 f -   2 -   1  -8
 2  2
(b) ( g  f )(0)  g ( f (0))  g  2(0)  1
 g - 1  - 1  3  -
1 7
2 2

7. 1 
(c) ( g  g )(1)  g ( g (1))  g  (1)  3 
2 
 5 1 5 5 17
 g -    -   3  -  3  -
 2 2 2 4 4
(d ) ( f  f )(2)  f ( f (2))  f  2(2)  1
 f  4  1
 f 4  1
 f 3  23  1  6  1  6  1  5