Composite Functions

Example 1: Given functions f(x) = x2 – 3x and g(x) = 2x + 1, find f g.

The notation "f g" means f (g(x)). In other words, replace x in function f with 2x + 1 (the g function).

f (x ) = (x)2 – 3(x)

Composite Functions

Example 1: Given functions f(x) = x2 – 3x and g(x) = 2x + 1, find f g.

The notation "f g" means f (g(x)). In other words, replace x in function f with 2x + 1 (the g function).

f (x ) = ( )2 – 3( )

f g = f (g(x)) = (2x + 1)2 – 3(2x + 1)

f g = 4x2 + 4x + 1 – 6x – 3

f g = 4x2 – 2x – 2

Composite Functions

2)(

xx

xfExample 2: Given functions and

find f g and state its domain.

,1

1)(

xxg

2gf

x

x 21

11

1

x

x

Simplify the complex fraction by multiplying the numerator and denominator by x + 1.

Composite Functions

121

11

11

11

1

x

xx

xx

gf

,221

1

x

gf 32

1

x

gf

At first glance it might appear that the domain of f g is the set of all real numbers except - 1.5.

Composite Functions

321

x

gf

Since the g function is not defined for x = - 1, neither is the f g function.

However, remember that the g function,

replaced x in the f function.

,1

1

x

xg

Therefore, the domain of f g is:

( - , - 1.5 ) ( - 1.5, - 1 ) ( - 1, ).

Composite Functions

Try: Given functions and

find f g and state its domain.

,5)( xxg5)( 2 xxf

Its domain is: [ - 5, ).

The composite function, f g = x.

Composite Functions