Operations of Functions
and Composite Functions
How functions operate
Overview
𝑓 𝑥
 𝑓+𝑔 𝑥
 𝑓−𝑔 𝑥
 𝑓·𝑔 𝑥
 𝑓/𝑔 𝑥
 𝑓∘𝑔 𝑥
 𝑓∘𝑓 𝑥
or
𝑓
𝑔
𝑥 or
𝑓÷𝑔 𝑥
𝑓(𝑥)
 This is read “f of x”
 For example when we see:
𝑓 𝑥 = 2𝑥 3 − 3𝑥 2 + 𝑥 − 4
 Which means function It means whatever we take the “f” of,
we need to:
of x
 It is usually followed by
instructions of what to
do with the input (x)
 Take 2 times the input cubed
 Minus 3 times input squared
 Add the input
 Then subtract 4
 Other common forms
 More simply put, if -2 is the input:
are 𝑔 𝑥 𝑎𝑛𝑑 ℎ 𝑥
3
2
𝑓 −2 = 2(−2) − 3 −2
= −34
+ (−2) − 4
𝑓+𝑔 𝑥
 𝑓 + 𝑔 𝑥 is just like
saying 𝑓 𝑥 + 𝑔(𝑥)
Just take one
function, put a plus
sign and add the
second function
For example:
𝑓 𝑥 = 5𝑥 2 + 9
𝑔 𝑥 = 2𝑥 + 10
𝑓 + 𝑔 𝑥 → 𝑓 𝑥 + 𝑔(𝑥)
↓
5𝑥 2 + 9 + 2𝑥 + 10
↓
5𝑥 2 + 2𝑥 + 19
𝑓−𝑔 𝑥
 𝑓 − 𝑔 𝑥 is just like
saying 𝑓 𝑥 − 𝑔(𝑥)
 Just write the first
function, put a minus sign
and place the second
function in parentheses
after the minus sign
 Remember to distribute
the minus sign
 For example:
𝑓 𝑥 = 5𝑥 2 + 9
𝑔 𝑥 = 2𝑥 + 10
𝑓 − 𝑔 𝑥 → 𝑓 𝑥 − 𝑔(𝑥)
↓
5𝑥 2 + 9 − 2𝑥 + 10
↓
5𝑥 2 + 9 − 2𝑥 − 10
↓
5𝑥 2 − 2𝑥 − 1
𝑓·𝑔 𝑥
 𝑓 ∙ 𝑔 𝑥 is just like saying
𝑓 𝑥 × 𝑔(𝑥)
 Just write the first function
in parentheses, place the
second function in
parentheses after the first
function.
 Remember to distribute
the terms
 For example:
𝑓 𝑥 = 5𝑥 2 + 9
𝑔 𝑥 = 2𝑥 + 10
𝑓 ∙ 𝑔 𝑥 → 𝑓 𝑥 × 𝑔(𝑥)
↓
5𝑥 2 + 9 2𝑥 + 10
↓
10𝑥 3 + 50𝑥 2 + 18𝑥 + 90
𝑓/𝑔 𝑥
 𝑓/𝑔 𝑥 is just like saying
𝑓 𝑥 ÷ 𝑔(𝑥)
 Just write the first function
in the numerator, and
place the second
function in the
denominator
 (reduce if possible)
For example:
𝑓 𝑥 = 5𝑥 2 + 9
𝑔 𝑥 = 2𝑥 + 10
𝑓 𝑥
𝑓/𝑔 𝑥 →
𝑔(𝑥)
↓
5𝑥 2 + 9
2𝑥 + 10
𝑓∘𝑔 𝑥
 𝑓 ∘ 𝑔 𝑥 is just like saying
𝑓(𝑔 𝑥 )
 Just write the second
function in place of x
everywhere in the first
function
 (simplify if possible)
For example:
𝑓 𝑥 = 5𝑥 2 + 9
𝑔 𝑥 = 2𝑥 + 10
𝑓 ∘ 𝑔 𝑥 → 𝑓(𝑔 𝑥 )
↓
2
5(2𝑥 + 10) +9
↓
5 4𝑥 2 + 40𝑥 + 100 + 9
𝑓∘𝑓 𝑥
 𝑓 ∘ 𝑓 𝑥 is just like saying
𝑓(𝑓 𝑥 )
 Very similar to 𝑓 ∘ 𝑔 𝑥 ,
but instead of 2 different
functions:
 Just write the function in
place of x everywhere in
the first function
 (simplify if possible)
For example:
𝑓 𝑥 = 5𝑥 2 + 9
𝑓 ∘ 𝑓 𝑥 → 𝑓(𝑓 𝑥 )
↓
2
2
5(5𝑥 + 9 ) +9
↓
5 25𝑥 4 + 90𝑥 2 + 81 + 9