Operations on Functions

Lesson 3.5

Sums and Differences of Functions

• If f(x) = 3x + 7 and g(x) = x2 – 5 then,

• h(x) = f(x) + g(x) = 3x + 7 + (x2 – 5)

• So, h(x) = x2 + 3x – 2

• Now, if h(x) = f(x) – g(x), then

• 3x + 7 – (x2 – 5) = -x2 + 3x + 12

Example2( ) 1 ( ) 1,

, , !

For f x x and g x x find

f g f g and thedomainof both

Products and Quotients 2( ) 2 ( ) 4,

, , !

For f x x and g x x find

ffg and thedomainof both

g

If f(x) = 3x2 + 7 and g(x) = 4, then f(x)g(x) = 4(3x2 + 7) = 12x2 + 28

Composition of Functions

• If f and g are functions, then the composite function of f and g is (g◦f)(x) = g(f(x))

• The expression g ◦ f is read g circle f or f followed by g. The functions are applied right to left.

Examples2 1

( ) 4 1, ( ) ,2

:

. ( )(2)

. ( )( 1)

. ( )( )

. ( )( )

If f x x and g xx

find the following

a g f

b f g

c g f x

d f g x

Domain of g ◦ f• Let f and g be functions. The domain of

g ◦ f is the set of all real numbers x such that– x is in the domain of f– f(x) is in the domain of g

Example2( ) 2 ( ) 1

:

.

.

If f x x and g x x

find

a g f and f g

b find thedomainof eachcomposite function

Writing a Function as a Composite2( ) 3 1

.

Write h x x as a compositionof functions in

twodifferent ways

Applications

3

A cylindrical container is being filled with water.

After t minutes, the height of the water in the container

1is h(t) = 3 inches. The volume V of the water

2

in the container is given by V(h) = .4

E

t

h

xpress the volume as a function of time by finding

(V h)(t) = V(h(t)). and compute the volume at

t = 2 minutes.