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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