1
U2L1
4.1: Composite Functions
Learning Target: I can: (1) evaluate a composite function (2) find the domain of a composite function
Exploration: Form a Composite Function
In the mail you receive a coupon for $5 off of a pair of jeans. When you arrive at
the store, you find that all jeans are 25% off. You find a pair of jeans for $55.
(a) If you use the $5 off coupon first, and then you use the 25% off on the remaining amount,
how much will the jeans cost?
(b) If you use the 25% off first, and then you use the $5 off on the remaining amount, how
much will the jeans cost?
(c) Would you prefer to use the $5 off coupon and then the 25% off or the 25% off and then
the $5 coupon? Justify your response.
( d) Now consider the situation if the jeans cost x dollars. Write a function 𝒇(𝒙) that
represents the cost of the jeans after the $5 off coupon.
(e) If the jeans cost x dollars write a function 𝒈(𝒙) that represents the cost of the jeans after
the 25% discount.
(f) Write a new function 𝒔(𝒙) that represents the cost of the jeans if the $5 off coupon is
applied first and the 25% discount is applied second. Using a composition of functions, we
can write this in terms of f ( x) and g ( x) as follows:____________________
(g) Write a new function 𝒓(𝒙) that represents the cost of the jeans if the 25% discount is
applied first and the $5 off coupon is applied second. Using a composition of functions, we
can write this in terms of f ( x) and g ( x) as follows:____________________
(h) Now, if x= 55, put it in 𝒔(𝒙) and 𝒓(𝒙). Do you get the same answer as you did in part (a)
and (b), respectively?
2
U2L1
Composite functions: When the _________________ of one function becomes the __________________
of another function
Given two functions f and g, the composite function, denoted by _______________________
(read as “f composed with g” or “ f of g “ ) is defined by_______________________________.
Note: f g does not mean f multiplied by g(x). It means input the function g into the
( ( ))
function f: f o g = f g x ¹ f (x)g(x)
Example 1: Evaluate a Composite Function
Suppose that f  x   3x2  1 and g  x   3 x . Find:
(a) ( f g )( x)
(b) ( f g )(1)
(c) ( g f )( x)
(d) ( g f )(1)
3
U2L1
Group Work 1!: Evaluate a Composite Function
Suppose m( x)  3x  5 and n( x)  2 x2  1. Find:
(a)  m n  ( x )
(b)
(𝑛 ∘ 𝑚)(2)
(c)  m m  ( x )
Example 2: Evaluate a Composite Function Using a Table
If f (x) and g(x) are polynomial functions, use the table of values for f (x) and g(x) to complete
the table of values for ( f g )( x) .
x
g  x
x
-2
-1
0
1
2
4
1
0
1
4
0
1
2
3
4
f  x
3
4
5
6
7
Example 3: Evaluate a Composite Function Using a Graph
Use the given graphs of f and g to approximate each
expression:
(a) f ( g (2))
(b) g ( f (1))
(c) f ( f (1))
(d) g ( g (1))
(e) ( f g )(6)
(f) ( g f )(3)
x
-2
-1
0
1
3
( f g )( x)
4
U2L1
Example 4: Find the Components of a Composite Function
Find functions f and g such that f g  H if:
(a) H(x) = x + 3
(b) H  x  
( c ) 𝐻(𝑥) = (5𝑥 − 8)6
(d ) 𝐻(𝑥) =
1
2x  3
2
1
𝑥 3 −7𝑥+2
U2L1
How do you find the domain of composite functions?
Example 5: Find the Domain of a Composite Function
1
4
( a ) Let f  x  
. Find ( f g )( x) and state the domain.
and g  x  
x4
x2
5
6
U2L1
Group Work 3!
1. Suppose that f  x  
1
and g  x   x  1 . Find and fully simplify the following and
x
state the domain.
(a) ( f g )( x)
(b) ( f
f )( x)
2𝑥
2. Let 𝑓(𝑥) = 5𝑥 − 7 , 𝑔(𝑥) = 𝑥−3 and ℎ(𝑥) = √4𝑥 + 8. Find and fully simplify the
following and state the domain.
(a) ( g f )( x)
(b) (𝑓 ∘ ℎ)(𝑥)