Operations on Functions and Composition Notes

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