### Section 1.8 - Larson Precalculus

```Section 1.8
Combinations of Functions
Section 1.8 Combinations of Functions: Composite Functions
Objective: In this lesson you learned how to find arithmetic
combinations and compositions of functions.
Course Number
I. Arithmetic Combinations of Functions (Pages 76−77)
Date
Instructor
Just as two real numbers can be combined with arithmetic
operations, two functions can be combined by the operations of
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a
to create new functions. A combined function like this is called
What you should learn
How to add, subtract,
multiply, and divide
functions
an arithmetic combination of functions.
The domain of an arithmetic combination of functions f and g
consists of . . .
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aaaaa aa aaa aaaaaaa aaaaaaaaaaa aaaa aaaa a aa
Let f and g be two functions with overlapping domains.
Complete the following arithmetic combinations of f and g for all
x common to both domains:
1) Sum: ( f + g )( x ) =
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a
2) Difference:
3) Product:
4) Quotient:
Example:
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( f − g )( x ) =
( fg )( x) =
f
 (x) =
g
a
aaaa a aaaa
aaaa aaaa aaaa a a
a
Let f ( x ) = 7 x − 5 and g ( x ) = 3 − 2 x . Find
( f − g )(4) .
aa
II. Composition of Functions (Pages 78−79)
The composition of the function f with the function g is defined
as ( f o g )( x ) =
a
aaaaaaa
.
What you should learn
How to find the
composition of one
function with another
function
Chapter 1
Functions and Their Graphs
For the composition of the function f with g, the domain of
( f o g ) is . . .
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aaaa aa aa aaa aaaaaa aa aa
For two functions f and g, to find ( f o g )(x ) , . . .
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aaaaaaaaaa aa a aa a aaaa aaa aaaaaaaaa aaaaaaaaaa aaaaa
aaaaaaa aaaaa aaaa aaaaaaaaa
Example:
Let f ( x ) = 3x + 4 and let g ( x) = 2 x 2 − 1 . Find
(a) ( f o g )(x ) and (b) ( g o f )(x ) .
aaa aa a a a
aaa aaa a a aaa a aa
III. Application (Page 80)
The function f ( x ) = 0.06 x represents the sales tax owed on a
purchase with a price tag of x dollars and the function
g ( x) = 0.75 x represents the sale price of an item with a price tag
of x dollars during a 25% off sale. Using one of the combinations
of functions discussed in this section, write the function that
represents the sales tax owed on an item with a price tag of x
dollars during a 25% off sale.
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