```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 () = 2 − 3 and () =  2 − 1:
 Sum of functions: ( + )() = () + ()
o () + () = (2 − 3) + ( 2 − 1) =  2 + 2 − 4
o Add the two functions together and combine like terms
 Difference of functions: ( − )() = () − ()
o () − () = (2 − 3) − ( 2 − 1) = − 2 + 2 − 2
o Subtract the two functions, REMEMBER TO PUT THE SECOND IN PARENTHESIS AND
DISTRIBUTE THE NEGATIVE, then combine like terms
 Product: ()() = () ∙ ()
o ()() = (2 − 3)( 2 − 1) = 2 3 − 3 2 − 2 + 3
o FOIL the two functions together

 Division:( ) () =
o

()
()
=
2−3
2 −1
()
()
,  ≠ ±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. () =  +  & () = √ −
B. () = √ −  & () =
a.) ( + )()
a.) ( + )()
b.) ( − )()
b.) ( − )()
c.) ()()
c.) ()()

d. ( ) () and find the domain

FROM A GRAPH:

d. ( ) () and find the domain

+
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Composition of Functions
The composition of the function f with the function g is:
( ∘ )() = (())
Which is read f of g of x. The domain of ( ∘ ) is the set of all x in the domain
of g such that g(x) is the domain of f.
Given () =  + 2 & () = 4 −  2 :
a.) ( ∘ )()
b.) ( ∘ )()
c.) ( ∘ )()
d.) ( ∘ )(−2)
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