Definition

f g

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

Suppose f and g are functions. Then the composite function can be described by the equation

Example

2 ( ) 3 1 ( ) 2 3,

( )( ) ( )( ).

If f x x x and g x x

Find f g x and g f x

( f g)(x) 4x2 18x 19

(g f )(x) g(x2 3x 1)

2( )( ) 2 6 1g f x x x

Another example

If f (x) x 2 and g(x) x2 3, find f (g(x))

and g( f (x)).

f (g(x)) f (x2 3)

x2 3 2

x2 1

Next

f (g(x)) f (x2 3)

x2 3 2

x2 1

Next

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

(x 2)2 3

x2 4x 1

Find f (g(x)) and g( f(x)).

a) f (x) 2x 7 ; g(x) 3x 1

b) f (x) x2 2x ; g(x) x 3

Answers:

a) ( f g)(x) 6x 2

b) (g f )(x) x2 2x 3

Now we will go over how to find a value of composite of functions.

If f (x) x2 2x 1 ; g(x) 2x 3 ; h(x) 4x

Find each value.

) ( )( 2)a f g

Next

) ( )( 2)b g f

) ( )(3)

)( )( 1)

c h f

d g h

) ( )( 2) 1b g f

) ( )( 2) 64a f g

) ( )(3) 64

) ( )( 1) 5

c h f

d g h