2.6 – Combinations of Functions
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

We can add, subtract, multiply and divide functions.
We can find the composition of one function with another and use them to model and solve real-life problems.
Arithmetic Combinations
Using 𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) = 𝑥 2 − 1:
 Sum of functions: (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)
o 𝑓(𝑥) + 𝑔(𝑥) = (2𝑥 − 3) + (𝑥 2 − 1) = 𝑥 2 + 2𝑥 − 4
o Add the two functions together and combine like terms
 Difference of functions: (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)
o 𝑓(𝑥) − 𝑔(𝑥) = (2𝑥 − 3) − (𝑥 2 − 1) = −𝑥 2 + 2𝑥 − 2
o Subtract the two functions, REMEMBER TO PUT THE SECOND IN PARENTHESIS AND
DISTRIBUTE THE NEGATIVE, then combine like terms
 Product: (𝑓𝑔)(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥)
o (𝑓𝑔)(𝑥) = (2𝑥 − 3)(𝑥 2 − 1) = 2𝑥 3 − 3𝑥 2 − 2𝑥 + 3
o FOIL the two functions together
𝑓
 Division:( ) (𝑥) =
o
𝑔
𝑓(𝑥)
𝑔(𝑥)
=
2𝑥−3
𝑥 2 −1
𝑓(𝑥)
𝑔(𝑥)
, 𝑥 ≠ ±1
o Write the ratio of the two functions as a fraction
o Remember to inspect the domain of each to find the excluded x-values
A. 𝒇(𝒙) = 𝒙𝟐 + 𝟔 & 𝒈(𝒙) = √𝟏 − 𝒙
B. 𝒇(𝒙) = √𝒙𝟐 − 𝟒 & 𝒈(𝒙) =
a.) (𝑓 + 𝑔)(𝑥)
a.) (𝑓 + 𝑔)(𝑥)
b.) (𝑓 − 𝑔)(𝑥)
b.) (𝑓 − 𝑔)(𝑥)
c.) (𝑓𝑔)(𝑥)
c.) (𝑓𝑔)(𝑥)
𝑓
d. ( ) (𝑥) and find the domain
𝑔
FROM A GRAPH:
𝑓
d. ( ) (𝑥) and find the domain
𝑔
𝒙𝟐
𝒙𝟐 +𝟏

Composition of Functions
The composition of the function f with the function g is:
(𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥))
Which is read f of g of x. The domain of (𝑓 ∘ 𝑔) is the set of all x in the domain
of g such that g(x) is the domain of f.
Given 𝑓(𝑥) = 𝑥 + 2 & 𝑔(𝑥) = 4 − 𝑥 2 :
a.) (𝑓 ∘ 𝑔)(𝑥)
b.) (𝑔 ∘ 𝑓)(𝑥)
c.) (𝑓 ∘ 𝑓)(𝑥)
d.) (𝑔 ∘ 𝑓)(−2)