* 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.
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
= − = −
