table of contents composite functions example 1:given functions f(x) = x 2 – 3x and g(x) = 2x + 1,...
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Table of Contents
Composite Functions
Example 1: Given functions f(x) = x2 – 3x and g(x) = 2x + 1, find f g.
The notation "f g" means f (g(x)). In other words, replace x in function f with 2x + 1 (the g function).
f g = f (g(x)) = (x)2 – 3(x)
f g = 4x2 + 4x + 1 – 6x – 3
f g = 4x2 – 2x – 2
f g = f (g(x)) = (2x + 1)2 – 3(2x + 1)
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Composite Functions
Slide 2
2)(
xx
xfExample 2: Given functions and
find f g and state its domain.,1
1)(
xxg
2gf
x
x 21
11
1
x
x
Simplify the complex fraction by multiplying the numerator and denominator by x + 1.
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Composite Functions
Slide 3
121
11
11
11
1
x
xx
xx
gf
,221
1
x
gf 32
1
x
gf
At first glance it might appear that the domain of f g is the set of all real numbers except - 1.5.
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Composite Functions
Slide 4
321
x
gf
Since the g function is not defined for x = - 1, neither is the f g function.
However, remember that the g function,
replaced x in the f function.
,1
1
x
xg
Therefore, the domain of f g is:
( - , - 1.5 ) ( - 1.5, - 1 ) ( - 1, ).
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Composite Functions
Slide 5
Try: Given functions and
find f g and state its domain.
,5)( xxg5)( 2 xxf
Its domain is: [ - 5, ).
The composite function, f g = x.
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Composite Functions