with

Function Operations

• Addition:

• Subtraction:

• Multiplication:

• Division:

)()())(( xgxfxgf

)()())(( xgxfxgf

)()())(( xgxfxgf

0)(,)(

)()(

xg

xg

xfx

g

f

Examples:If and 1) Find (f + g)(x)

2) Find (f - g)(x)

3) Find

4) Find

))(( xgf

132)( 2 xxxf xxg 5)(

)(xg

f

Examples:If and 132)( 2 xxxf xxg 5)(

1) Find (f + g)(3)

2) Find (f - g)(0)

3) Find

4) Find

)2)(( gf

)3(

g

f

OF

Definition of composite functions

Suppose f and g are functions such that the range (output) of g is the subset of the domain (input) of f. Then the composite function

f g

can be described by the equation

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

Let’s do an example together.

EX: If f (x) x2 3x 1 and g(x) 2x 3,

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

( f g)(x) f (2x 3)

f (2x 3) (2x 3)2 3(2x 3) 1

• Substitute 2x-3 in for g(x):

• Substitute into f(x):

Next

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

Simplify:

f (2x 3) (2x 3)2 3(2x 3) 1

4x2 12x 9 6x 9 1

4x2 18x 19

Now lets find (g f )(x)

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

g(x 2 3x 1) 2(x2 3x 1) 3• Substitute f(x) into g(x):

(g f )(x) 2x2 6x 2 3

2x2 6x 1

• Substitute for g(x):x2 3x 1

• Simplify:

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

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

(x 2)2 3

x2 4x 1

Try these on your own: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.

a) ( f g)( 2) f ( 2x 3) Substitute in g(x).

f ( 2( 2) 3) Substitute into g(x).

f (7) SimplifyNext

f (7) 72 2(7) 1 Substitute into f(x).

( f g)( 2) 64 Simplify

b) (g f )( 2) g(( 2)2 2( 2) 1)

g(1)

2(1) 3

1

Now you find the values using the same directions as in the last examples.

1) (h f )(3)

2)(g h)( 1)

Answer: 64

Answer: -5