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Pre-Calculus12
10.3 Composition of Functions
Warmup:
Given: f ( x) = 2 x + 5 and g ( x) = 3x − 4 find the following:
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a)
f (3)
b) f (−5)
c) f (a )
d) f (3 x − 4)
What would f ( g ( x)) be?
Composite Functions are functions that are formed from two other functions where the output of one function is
the input to the other function.
Consider: Evaluate g (7) , and then use this value as the input to f ( x).
We could write this as:
Composition is a new operation for functions and can be written:
Ex. #1 Given: f ( x) = 2 x + 3 and g ( x) = x 2 − 1 , find:
a) ( f g )( x)
b)
( f g )(3)
c) ( g f )( x)
d)
( g f )(3)
NOTE:
Ex. #2 State the domain and range of:
a)
f ( g ( x))
b) g ( f ( x))
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Ex. #3 Consider: f ( x) = x + 1 and g ( x) = x + 2 , determine the following and complete the table below.
a) ( f g )(3) .
b)
( g f )(−5)
c)
( g f )(−1)
d) m( x) = ( f g )( x)
e)
n( x) = ( g f )( x)
f)
g ( g ( x))
f ( x)
g ( x)
m( x) = ( f g )( x)
n( x) = ( g f )( x)
Domain
Range
Ex. #5 Determine two functions f ( x) and g ( x) where h( x) = ( f g )( x).
a) h( x) = 5 x − 10
b)
h( x) = ( x + 1) 2 − 7( x + 1) + 5
Ex. #6 Given the graph below. Sketch the graph of m( x) = f ( g ( x)) .
y = f ( x)
y = g ( x)
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