Section 2.7Combining Functions

Objectives:•To add, subtract, multiply and divide functions.•Composition of functions.

Two functions f and g can be combined to form new functions

f + g, f – g, fg, f/g

in a manner similar to the way we add, subtract, multiply, and divide real numbers.

( )( ) ( ) ( ) Domain ( )( ) ( ) ( ) Domain

( )( ) ( ) ( ) Domain

( )( ) Domain { | ( ) 0}( )

f g x f x g x A Bf g x f x g x A B

fg x f x g x A B

f f xx x A B g xg g x

Let f and g be functions with domains A and B.Then, the functions f + g, f – g, fg, and f/g

are defined as follows.

Ex 1. Let and

(a) Find the functions f + g, f – g, fg, and f/g and their domains.

(b) Find (f + g)(-1), (f – g)(3), (fg)(4), and (f/g)(2).

2( ) 4f x x x 2( ) 6g x x

Class Work

1. Let and

a.) Find the function and its domain.

b.) Find

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

f g

( )(8)f g

• Given two functions f and g, the composite function f ◦ g (also called the composition of f and g) is defined by:

(f ◦ g)(x) = f [g(x)]

Ex 2. Let f(x) = x2 and g(x) = x – 3

(a) Find the functions f ◦ g and g ◦ f and their domains.

(b) Find (f ◦ g)(5) and (g ◦ f )(7).

Class Work

2. Let f(x) = x2 and g(x) = x + 5.

(a) Find the functions f ◦ g and g ◦ f and their domains.

(b) Find (f ◦ g)(-2) and (g ◦ f )(4).

Ex 3.

If f(x) = and g(x) = , find the following functions and their domains.

(a) f ◦ g(b) g ◦ f(c) f ◦ f(d) g ◦ g

x 2 x

Class Work

3. Let . Find:a)

b)

c)

10, ( ) , ( ) 31xf x g x x h x xx

f g

g h

f h

HW #7 p219 1-9odd, 17-21 odd, 23-35 odd