8-7
Composition of Functions
1
1
Composition of Functions
Composite - formed by combining functions.
The output from one function becomes the
input for another function.
Written as g(f(x)) or [g  f ](x).
Evaluate the inside function, f(x), first.
Then use the output for g(x).
2
f ( x)  3x  2
Find: 1. g(f(-2))
g ( x)  x  2
2
2. [ f  g ](-2)
f(-2) = 3(-2) - 2 [ f  g ](-2) = f(g(-2))
g(-2) = (-2)2 - 2
f(-2) = -8
g(-2) = 2
g(f(-2)) = g(-8)
g(f(-2)) = (-8)2 - 2
g(f(-2)) = 62
f(g(-2)) = 3(2) - 2
[ f  g ](-2) = 4
3
f ( x)  3x  2
Find: 1. g(f(x))
g ( x)  x  2
2
2. [ f  g ](x)
[ f  g ](x) = f(g(x))
f(x) = 3x - 2
g(f(x)) = g(3x - 2)
g(3x - 2) = (3x - 2)2 - 2
g(x) = x2 - 2
f(x2 – 2) = 3(x2 – 2) – 2
f(x2 – 2) = 3x2 – 6 – 2
g(3x - 2) = (9x2 – 12x + 4) - 2
g(f(x)) = 9x2 – 12x + 2
[ f  g ](x) = 3x2 – 8
4
h( x)  2 | x  5 | 8
Find: 1. h(k(2))
k(2) = 3(2) – 2(2)2
k(2) = 6 – 8
k(2) = -2
h(-2) = 2 |(-2) – 5| - 8
h(-2) = 2 |-7| - 8
h(-2) = 14 - 8
h(k(2)) = 6
k ( x)  3x  2 x 2
2. [h  k](3)
[h  k](3) = h(k(3))
k(3) = 3(3) – 2(3)2
k(3) = -9
h(-9) = 2 |(-9) – 5| - 8
[h  k](3) = 20
5
p( x)  3x  8
Find: 1. 2p(x) + q(x)
2(3x + 8) + (2x – 12)
6x + 16 + 2x – 12
8x + 4
q ( x )  2 x  12
2. 3p(q(x))
3[3(2x – 12) + 8)]
3[6x – 36 + 8]
3[6x – 28)]
18x – 84
6
f ( x)  2 x  4 x
2
g ( x)  1  3x
Find the following:
1. f ( g ( x))
f ( x)
2.
 3 g ( g ( x))
2
2(1  3 x)  4(1  3 x)
2
2
x

4
x
2
 3[1  3(1  3 x)]
2(1  6 x  9 x )  4  12 x
2
2
2  12 x  18 x 2  4  12 x
x  2 x  3  9(1  3 x)
18 x 2  2
2
x  2 x  3  9  27 x
2
x  29 x  6
2
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f ( x)  2 x  4 x
2
g ( x)  1  3x
Solve for x:
1. f ( x)  g ( x)
2 x 2  4 x  1  3x
2
2x  x 1  0
 2 x  1 x  1  0
 1 
x   ,1
 2 
2. 2( g ( x))  f ( 1)
2 1  3 x   2(1)  4(1)
2
2  6x  2  4
2  6x  6
6 x  4
 2
x   
 3
8
Finite Sets
Finite Sets: Given 2 functions: f and g,
[f  g](x) and [g  f](x) only exist if the
range of each function is a subset of the
domain of the other function.
If the output from the first function is not
an input for the second function, the
composition does not exist (DNE).
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Find [g  f](x) if it exists.
f = {(-3, 1), (-1, 2), (1, 3), (3, 4)}
and
g = {(4, -3), (3, 1), (2, 3), (1, 5)}
(g  f ) = {(-3,5), (-1,3), (1,1), (3,-3) }
10
Find [f o g] if it exists.
f = {(-3, 1), (-1, 2), (1, 3), (3, 4)}
and
g = {(4, -3), (3, 1), (2, 3), (1, 5)}
(f  g ) = {(4,1), (3,3), (2,4), (1,?) }
There is no 5 in the domain of f
(f  g ) DNE
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Use f and g to find the following:
f(x)
1.
f ( g ( f (7)))
Start inside
f (7) = 2
g (2) = 4
f (4) = 3
g   2, 4  , 1,5 ,  3,0  6,1
2.
f
g g  6 
Start at right
g (6) = 1
g (1) = 5
f (5) = 4
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f  1, 3 ,  2, 4  ,  3, 1 ,  4, 2 
g   3, 2  ,  2, 0  ,  4,3  ,  1,1
h(x)
Find (f  g) and (g  f ) if they exist.
(f  g) = {(-5,4), (2,? DNE
(g  f ) = {(1,2), (2,3), (3,1), (4,0)}
Find (g  h  f )(2)
Find (f(g(h(6)))
f(2) = 4
h(6) = -1
h(4) = 2
g(-1) = 1
f (1) = -3
g(2) = 0
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