This document covers the composition of functions, focusing on combining functions using algebraic operations, creating, evaluating, and decomposing composite functions. Key concepts include defining composite functions, finding their domains, and examples illustrating these processes. It also includes exercises and classwork for practice.

1. 3.4 Composition of Functions
Chapter 3 Functions

2. Concepts and Objectives
⚫ Objectives for this section are:
⚫ Combine functions using algebraic operations.
⚫ Create a new function by composition of functions.
⚫ Evaluate composite functions.
⚫ Find the domain of a composite function.
⚫ Decompose a composite function into its component
functions.

3. Operations on Functions
⚫ Given two functions f and g, then for all values of x for
which both f(x) and g(x) are defined, we can also define
the following:
⚫ Sum
⚫ Difference
⚫ Product
⚫ Quotient
( )( ) ( ) ( )
f g x f x g x
+ = +
( )( ) ( ) ( )
f g x f x g x
− = −
( )( ) ( ) ( )
fg x f x g x
= 
( )
( )
( )
( )
, 0
f x
f
x g x
g g x
 
= 
 
 

4. Operations on Functions (cont.)
⚫ Example: Let and . Find each
of the following:
a)
b)
c)
d)
( ) 2
1
f x x
= + ( ) 3 5
g x x
= +
( )( )
1
f g
+ ( ) ( )
1 1
g
f
= + ( )
2
5
1 1
1 3
= + +
+ 0
2 1
8
= + =
( )( )
3
f g
− − ( ) ( )
2
3 5
3 3
1  
− +
 
−
− +
= ( ) 4
10 1
4
−
= − =
( )( )
5
fg ( ) ( )
2
3 5 5
5 1  
+
 
+
= ( )( ) 0
20
26 52
= =
( )
0
f
g
 
 
  ( )
2
5
0 1
3 0
+
+
=
5
1
=

5. Operations on Functions (cont.)
⚫ Example: Let and . Find
each of the following:
a)
b)
c)
d)
( ) 8 9
f x x
= − ( ) 2 1
g x x
= −
( )( )
f g x
+ 8 9 2 1
x x
= − + −
( )( )
f g x
− 8 9 2 1
x x
= − − −
( )( )
fg x ( )
8 9 2 1
x x
= − −
( )
f
x
g
 
 
 
8 9
2 1
x
x
−
=
−

6. Operations on Functions (cont.)
⚫ Example: Let and . Find
each of the following:
e) What restrictions are on the domain?
⚫ There are two cases that need restrictions: taking the
square root of a negative number and dividing by zero.
⚫ We address these by making sure the inside of g(x) > 0:
( ) 8 9
f x x
= − ( ) 2 1
g x x
= −
2 1 0
2 1
1
2
x
x
x
− 


So the domain must be
1 1
or ,
2 2
x
 
 
 
 

7. Composition of Functions
⚫ If f and g are functions, then the composite function, or
composition, of g and f is defined by
⚫ The domain of g ∘ f is the set of all numbers x in the
domain of f such that f(x) is in the domain of g.
⚫ So, what does this mean?
( )( ) ( )
( )
g f x g f x
=

8. Composition (cont.)
⚫ Example: A $40 pair of jeans is on sale for 25% off. If
you purchase the jeans before noon, the store offers an
additional 10% off. What is the final sales price of the
jeans?
We can’t just add 25% and 10% and get 35%. When
it says “additional 10%”, it means 10% off the
discounted price. So, it would be
( )
( )
25% off: .75 40 $30
10% off: .90 30 $27
=
=

9. Evaluating Composite Functions
⚫ Example: Let and .
(a) Find (b) Find
( ) 2 1
f x x
= − ( )
4
1
g x
x
=
−
( )( )
2
f g ( )( )
3
g f −

10. Evaluating Composite Functions
⚫ Example: Let and .
(a) Find (b) Find
(a)
( ) 2 1
f x x
= − ( )
4
1
g x
x
=
−
( )( )
2
f g ( )( )
3
g f −
( )( ) ( )
( )
2 2
f g f g
=
( )
4 4
4
2 1 1
f f f
   
= = =
   
−
   
( )
2 4 1 8 1 7
= − = − =

11. Evaluating Composite Functions
⚫ Example: Let and .
(a) Find (b) Find
(b)
( ) 2 1
f x x
= − ( )
4
1
g x
x
=
−
( )( )
2
f g ( )( )
3
g f −
( )( ) ( )
( )
3 3
g f g f
− = −
( )
( ) ( ) ( )
2 3 1 6 1 7
g g g
= − − = − − = −
4 4 1
7 1 8 2
= = = −
− − −

12. Composites and Domains
⚫ Given that and , find
(a) and its domain
The domain of f is the set of all nonnegative real
number, [0, ∞), so the domain of the composite
function is defined where g ≥ 0, thus
( )
f x x
= ( ) 4 2
g x x
= +
( )( )
f g x
( )( ) ( )
( ) ( )
4 2
f g x f g x f x
= = + 4 2
x
= +
4 2 0
x + 
1
2
x  −
1
so ,
2
 
− 

 

13. Composites and Domains
⚫ Given that and , find
(b) and its domain
The domain of f is the set of all nonnegative real
number, [0, ∞). Since the domain of g is the set of all
real numbers, the domain of the composite function
is also [0, ∞).
( )
f x x
= ( ) 4 2
g x x
= +
( )( )
g f x
( )( ) ( )
( ) ( )
g f x g f x g x
= = 4 2
x
= +

14. Composites and Domains (cont.)
⚫ Given that and , find
and its domain
( )
6
3
f x
x
=
−
( )
1
g x
x
=
( )( )
f g x
( )
( ) 1
f g x f
x
 
=  
 
6
1
3
x
=
−
6 6
1 3 1 3
x x
x x x
= =
−
−
6
1 3
x
x
=
−

15. Composites and Domains (cont.)
⚫ Given that and , find
The domain of g is all real numbers except 0, and the
domain of f is all real numbers except 3. The expression
for g(x), therefore, cannot equal 3:
( )
6
3
f x
x
=
−
( )
1
g x
x
=
1
3
x
=
1 3x
=
1
3
x =
( )
1 1
,0 0, ,
3 3
   
−   
   
   

16. Decomposition of Functions
⚫ In some cases, it is necessary to decompose a
complicated function. In other words, we can write it as
a composition of two simpler functions.
⚫ There may be more than one way to decompose a
composite function, so we may choose the
decomposition that appears to be the most expedient.

17. Decomposition of Functions
⚫ Example: Write as the composition of
two functions.
⚫ We are looking for two functions, g and h, so
f(x)=g(h(x)). To do this, we look for a function inside
a function in the formula for f(x).
⚫ As one possibility, we might notice that the
expression 5 ‒ x2 is inside the square root. We could
then decompose the function as
( ) 2
5
f x x
= −
( ) ( )
2
5 and
h x x g x x
= − =
( )
( ) ( )
2 2
5 5
g h x g x x
= − = −

18. Classwork
⚫ College Algebra 2e
⚫ 3.4: 6-16 (even); 3.3: 16-24 (even); 3.2: 38-54 (even)
⚫ 3.4 Classwork Check
⚫ Quiz 3.3