compositions notes
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Compositions Notes. Add earlier graphs where they have to say what is f(3)…. Function Compositions. Lecture: 2D Pre AP & GT Precalculus. Homework page 64 #88. Corners of equal size are cut from a square with sides of length 8 meters A. Write the area of the figure as a function of - PowerPoint PPT PresentationTRANSCRIPT
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Compositions Notes
• Add earlier graphs where they have to say what is f(3)…
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Function Compositions
Lecture: 2DPre AP & GT Precalculus
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Homework page 64 #88
• Corners of equal size are cut from a square with sides of length 8 meters
• A. Write the area of the figure as a function of Area Function:
Book Domain: (0,4) … Alternative
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2( ) 64 2A x x
(0,4 2)
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Homework page 64 #88
B.
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Book Range: (32,64) Alternative (0,64)
Sometimes there are multiple interpretations of anauthentic problem. Problems can have multiple solutions
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Homework page 64 #88
C. If the figure has no area
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If x=4, the figure is a square with side 4 2
4 2x
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Agenda• Combining Functions
• Function Compositions
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Function Algebra
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Example• Given find such that:
• a.
a: • b.
b: • c.
c:
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2( ) 1f x x ( )g x4( )* ( ) 1f x g x x
2( ) ( ) 3f x g x x
( ) 1
( ) 3
f x
g x
2( ) 1g x x
2( ) 2 1g x x
2( ) 3 3g x x
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Composition
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•We say, “f composed with g of x” OR “f of g of x”
• Range of g(x) is the domain of f(g(x))
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Examples• If and
then
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2( ) 2 1f x x x ( ) 2g x x
( ) ( ( ))f g x f g x
2( 2) ( 2) 2( 2) 1f x x x
( )f g x
2( ) 6 9f g x x x
( ) ( 2)f g x f x
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Graphical Example
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Based on the function to right, what is )))3((( fff
0)3( f
2)0())3(( fff
3)2())0(()))3((( ffffff
f g
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Example• If and
then
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2( ( )) 5 4f g x x x ( ) 2g x x
( )f x ( ( )) ( 2)f g x f x
2( 2) 5 4f x x x 2( 2 2) ( 2) 5( 2) 4f a a a
2( ) 2f a a a 2( ) 2f x x x
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Composition Graphically
• The point a is in the domain
The red line is the range of f(x)but is also the domain of g(f(x))
On a graph to change from domain to range shift left to y=x to find corresponding domain element of g(x)
Then find corresponding point in the range of g(x) and hence the range of g(f(x)).
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Reflection• Is for all functions, f
and g? No, this is not true…composition is not
commutative, order matters.
Counterexample:
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f g g f
2( )f x x( ) 2g x x2 2( ( )) (2 ) 4f g x x x 2 2( ( )) 2( ) 2g f x x x
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Show & Tell•
•
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Fractal Curves are made via a function composition process
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Composition Calisthenics
F G F o G
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2 2 1x x
1x 2 1x
1
2x
2 1x 2 2x
x2x
2x2 2 1x x 2 2x x
4 1x
4x
x2 1x
x
1x
2 x
Match across the columns
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Homework
• Pg 89 #6-12(even)Pg 90 #44-50 (even)
• Pg 91 #62
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