COMPOSITE FUNCTIONS Mr. Velazquez

Honors Precalculus

DOMAIN OF A FUNCTION

( )( )

( ) ( )

( ) ( ) ( )

2

2

3x+5g(x)=

x 4 5

x 4 5 0

5 1 0

5 0 1 0

x=5 x=-1

: - ,-1 1,5 5,

x

x

x x

x x

Domain

− −

− − =

− + =

− = + =

− )

( ) 2 5

2 5 0

2x 5

5 x

2

5: ,

2

h x x

x

Domain

= −

−

DOMAIN OF A FUNCTION

( )

2( ) 7

- ,

f x x x

Domain

= −

( )

( ) 4 5

: ,

f x x

Domain

= −

−

( )

3( )

: ,

f x x

Domain

=

−

Linear Function

Polynomial Function

Odd Root Function

All of these types

of functions have

a domain that

includes all the

real numbers

DOMAIN OF A FUNCTION

Find the domain of the function

4x-1f(x)=

3x+2

Find the domain of the function

f(x)= 4x-1

ALGEBRA OF FUNCTIONS

ALGEBRA OF FUNCTIONS

For two functions, 𝑓 𝑥 = 2𝑥 and 𝑔 𝑥 = 𝑥 − 1

If 𝑓

𝑔can be simplified, determine the domain before simplifying

ALGEBRA OF FUNCTIONSThe domain of f+g is the set of all real numbers that

are common to the domain of f and the domain of g.

Thus we must find the domains of f and g before

finding their intersection.

Suppose ( ) x+3 af x = nd g(x)= x-2 then

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

Now for their domains.

( ) x+3 g(x)= x-2

3 0 x-2 0

3 x 2

So the domain for the sum of the functions

is x 2 which in inter

x

f x

x

x

+ −

=

+

−

)val notation is 2,

ALGEBRA OF FUNCTIONS

− − − − − − − − −

−

−

−

−

−

−

−

x

y

)

The graph of (f+g)(x)= x+3 2

confirms that the domain of this function is 2,

x+ −

ALGEBRA OF FUNCTIONS2If f(x)=5x-1 g(x)=x 2 1 Find each of the following:

(f+g)(x)

(f-g)(x)

(fg)(x)

f( )

g

x

x

− +

ALGEBRA OF FUNCTIONS

1 1If f(x)= g(x)= Find the domain

2 1

of the following:

(fg)(x)

f( )

g

x x

x

−

ALGEBRA OF FUNCTIONS

If f(x)= x-1 g(x)= x-6 Find the domain for:

(f-g)(x)

COMPOSITION OF FUNCTIONS

COMPOSITION OF FUNCTIONS

COMPOSITION OF FUNCTIONS

( )( )

2Given f(x)= and g(x)= x.

x-3

a. Find f g b. Find the domain of f gx

COMPOSITION OF FUNCTIONS

( )( )

2

2

( ) 3 2 can be written as the

composition of what two functions?

g(x)= 3 2

( )

( )

h x x x

x x

f x x

h x f g x

= − +

− +

=

= ( )( )

2

2

( ) 3 2 can be written as the

composition of what two functions?

g(x)= 3 2

( )

( )

h x x x

x x

f x x

h x f g x

= − +

− +

=

=

COMPOSITION OF FUNCTIONS

2

Express h(x) as a composition of two functions:

1( )

9 64h x

x=

−

( )4

2

Express h(x) as a composition of two functions:

( ) 6 5 h x x x= − +

CLASSWORK & HOMEWORK

MATH JOURNAL: Summarize what you learned

CLASSWORK: COMPOSITE FUNCTIONS – Perform the given operations for the two functions below:

a) Find 𝑓 + 𝑔

b) Find the domain of 𝑓 and 𝑔

c) Find 𝑓𝑔 and its domain

d) Find 𝑓

𝑔and its domain

e) Find 𝑓 ∘ 𝑔 and its domain

𝒇 𝒙 = 𝒙𝟐 + 𝟏 𝒈 𝒙 = 𝟐 − 𝒙

Homework:

Functions HW 3 (Math XL)

Due 10/6