The document summarizes the concept of composition of functions. It defines composite functions as applying one function after another. To evaluate a composite function, work from the inside out by: 1) Finding the domain of the inner function 2) Applying the inner function 3) Using the result as the input for the outer function 4) The domain of the composite is the values that satisfy the domain of both functions.

1. Section 3-7
Composition of Functions

2. Composite g  f :
If f and g are functions, the the composite “g of f” is the
process of applying one function g after applying the other
function f.
For the domain, you need to compare the domains of the
two functions. First, look at the domain of the first
function (in this case f), and then the domain of the second.
( g  f ) ( x ) = g ( f ( x ))
Notation:

3. IMPORTANTE!!!
(That’s Spanish for “Important!!!”
Order matters when dealing with compositions. Work
from the inside out.
..
IT IS UBER IMPORTANT TO REMEMBER
THAT A VARIABLE IS JUST A PLACE
HOLDER FOR SOMETHING ELSE!!!

4. Example 1
Evaluate using the following functions:
() 1
()
f x =x 2
gx=
3x +1
2
⎛1⎞⎛1⎞
( ( )) =f⎜ =⎜
a. f g x ⎟ ⎝ 3x +1⎟
⎝ 3x +1⎠ ⎠
1 1
= =2
(3x +1)
2
9x + 6x +1

5. Example 1
Evaluate using the following functions:
() 1
()
f x =x 2
gx=
3x +1
⎛1⎞
( ( )) ⎛ 1⎞
b. f g 4 = f ⎜ ⎟ =f⎜ ⎟
()⎝ 13 ⎠
3 4 +1⎠
⎝
2
⎛ 1⎞ 1
=
=⎜ ⎟
⎝ 13 ⎠ 169

6. Example 1
Evaluate using the following functions:
() 1
()
f x =x 2
gx=
3x +1
) ( 4 ) = g ( f ( 4 )) = g ( 4 ) = g (16)
( 2
c. g  f
1 1
= =
()
3 16 +1 49

7. Domain of g: Domain of f:
D = {x: x = ℝ}
D = {x: x ≠ -1/3}
Here, even though f has a domain that contains all real
numbers, the composition f(g(x)) cannot contain -1/3 since
-1/3 is not in the domain of g. You finally need to look at
the domain of the final composite, as well, to find any
values that may not be in the domain.
( )
Domain of f  g :
D = {x: x ≠ -1/3}

8. Example 2
1
( )( )
4
( )( ) rh x =
h  r x = 2 −1 b.
a.
16x − 8x +1
2
x
( ( )) ( ( ))
1 4
c. r h 3 = d. h r −5 = −1
25
121

9. Example 2
e. Find the domain
1
( )( )
4
( )( ) rh x =
h  r x = 2 −1 b.
a.
16x − 8x +1
2
x
D = {x: x ≠ 0} D = {x: x ≠ 1/4}

10. Homework
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