Composite Functions in Mathematics
10.3 Composite Functions
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When two functions are applied in succession,
the resulting function is called the
composite
of the two given functions.
Composition of Two Functions
(
g
o
f
)(
x
) =
g
(
f
(
x
))
The range of
f
(
x
) becomes the
domain of
g
(
x
).
Evaluate
f
(
x
) then use the answer as
the input for
g
(
x
).
Notice: The function is evaluated
from the inside to the outside.
x is the input into the f function to get y
y is the input into the g function to get z
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10.3 Function
Composition
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Given
h
(
x
) = 4
x
+ 3
, determine the following:
b)
(
h
h
)(
-3
)
(
h
h
)(-3) =
h
(
h
(
-3
))
=
h
(4(
-3
) + 3)
=
h
(
-9
)
h
(
-9
) = 4(
-9
) + 3
=
-33
a)
(
h
h
)(
x
)
(
h
h
)
(
x
)
=
h
(
h
(
x
))
=
h
(4(
x
) + 3)
=
h
(
4
x
+ 3
)
h
(
4
x
+ 3
) = 4(
4
x
+ 3
) + 3
= 16
x
+ 12 + 3
=
16
x
+ 15
Note: (
h
h
)(
x
) ≠ (
hh
)(
x
)
(
h
h
)
(
x
)
=
h
(
x
)
x
h
(
x
)
Evaluating a Composition of a Functions
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Given
Determine an expression in simplest form for
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Using the function
determine the value of
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Determine the expressions for h(x)=
f
(g(
x
)) and k(x) = g(f(x))
The situation when
f
(g(
x
)) = g(f(x)) = x indicates that the original
functions are inverses.
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Use the graph to determine
the value of
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Page 507
1a,c, 2a,c, 3c, 4a,d,e, 5b,c, 6, 7, 8, 11, 13, 14, 15, 17
C1, C2
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Composite functions in mathematics involve applying two functions in succession, yielding a new function known as the composite. By evaluating functions in a specific order and considering their ranges and domains, composite functions provide a powerful tool for mathematical analysis.
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Presentation Transcript
10.3 Composite Functions = = + = + ( ) f x 4 ( ) g x 6 x x ( ) h x 5 x = ( ( )) f g x x ( x ) 6 4 6 x+ + = = = x ( ( ) g x ) 4 4 f How would you define composite functions? ( ) + 4 24 x Math30-1 1
Composition of Two Functions When two functions are applied in succession, the resulting function is called the composite of the two given functions. (g o f)(x) = g(f(x)) The range of f(x) becomes the domain of g(x). Evaluate f(x) then use the answer as the input for g(x). Notice: The function is evaluated from the inside to the outside. x is the input into the f function to get y y is the input into the g function to get z Math30-1 2
10.3 Function Composition 10.3 Function Composition = ( ) f x y ( ) = ( ) y f g x = ( ) g x y f ( ) = = ( ) y g f x ( ) f x y g Math30-1 3
= = + = + ( ) f x 4 ( ) g x 6 x x ( ) h x 5 x = ( ( )) h f x (4 ) h x ( ) = + 4 5 x 5 4 = + 4 5 x x , Math30-1 4
Evaluating a Composition of a Functions Given h(x) = 4x + 3, determine the following: a) (h h)(x) b) (h h)(-3) (h h)(x) = h(h(x)) = h(4(x) + 3) = h(4x + 3) (h h)(-3) = h(h(-3)) = h(4(-3) + 3) = h(-9) h(4x + 3) = 4(4x + 3) + 3 = 16x + 12 + 3 = 16x + 15 h(-9) = 4(-9) + 3 = -33 Note: (h h)(x) (hh)(x) (hh)(x) = h(x) xh(x) Math30-1 5
= = + ( ) h x 2 5 x 2 = ( ) g x 3 x ( ) f x 1 x Given Determine an expression in simplest form for ( 2 1 5 x ) = x ( ( )) h f x , 1 Why are there restrictions? ( ) = + ( )( ) ( ) k x 2 2 5 x = ( ) k x h g ( ( h f x ) ) ( = ( ) k x 1 ) h g ( x = ( ) k x 2 1 x x , 1 ) 2 = + ( ) k x 1 3 x ( ) = + ( ) k x 2 h x Math30-1 6
+ 2 ( ) 4 2 x x ( ) 4 f f = ( ) f x Using the function determine the value of f + 2 4 4 2 4 ( ) ( ) 4 = f f ( ) ( ) 4 ( ) = 10 f f f + 2 10 10 2 4 ( ) ( ) 4 = f f ( ) ( ) 4 = 13 f f Math30-1 7
= ( ) g x = ( ) f x log x 5x 5 Determine the expressions for h(x)= f(g(x)) and k(x) = g(f(x)) = = ( ) h x ( ( )) f g x ( ) k x ( ( )) g f x ( ) 5x ( ) h x = log ( ) k x = log x 5 5 5 = ( ) h x x = ( ) k x x The situation when f(g(x)) = g(f(x)) = x indicates that the original functions are inverses. Math30-1 8
Use the graph to determine the value of ( )(3) = = = + + (3) + (3) f g f g = ( ) f x y 0 ( 12) 12 ( ) ( ) 2 ( ) 4 = = g ( ) g x y g f = 12 Math30-1 9
Page 507 1a,c, 2a,c, 3c, 4a,d,e, 5b,c, 6, 7, 8, 11, 13, 14, 15, 17 C1, C2 Math30-1 10
