Compositions -...
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11/12/2013 1 Compositions • Composing functions mean to “implant one function into another” • “F of G” • “Evaluate G and put that value into F” • = ∘ • Graphs of composite functions are complex Compositions Compositions of inverse function must evaluate to x = = Consider the function = −4 and = 0. +2 () =2 0. +2 −4 = +4−4 =
Transcript of Compositions -...
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Compositions
• Composing functions mean to “implant one function into another”• “F of G”• “Evaluate G and put that value into F”• = ∘• Graphs of composite functions are complex
Compositions
Compositions of inverse function must evaluate to x= =Consider the function = − 4 and = 0. + 2( ) = 2 0. + 2 − 4= + 4 − 4=
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Important Note
The compositional operator (∘) is not the only operation that can bedone on functions. It is permitted to perform the operations of +,−,×÷ can be done with functions.
± = ± ( )= ( ) ( )÷ = ( )
Compositions
Given: = 2 − and = − 1Find:• ∘ 3• ( −2 )• Define the range of the graph of ∘
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Compositions
Given: = 2 − and = − 1Find: ∘ 3
Compositions
Given: = 2 − and = − 1Find: ∘ 3
1st: Evaluate = − = − =
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Compositions
Given: = 2 − and = − 1Find: ∘ 3
1st: Evaluate = − = − =Last: Evaluate = − = − =∘ =
Compositions
Given: = 2 − and = − 1Find: ( −2 )
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Compositions
Given: = 2 − and = − 1Find: ( −2 )
1st: Evalute − = − − = − − = −
Compositions
Given: = 2 − and = − 1Find: ( −2 )
1st: Evaluate − = − − = − − = −Last: Evaluate − = − − = − − = −− = −
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Compositions
Given: = 2 − and = − 1Find: Range of ∘
Compositions
Given: = 2 − and = − 1Find: Range of ∘Graph: = − − −
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Compositions
Given: = 2 − and = − 1Find: Range of ∘Graph: = − − −
: [− . , ∞)or∘ ≥ − .
Compositions
Given: = and =Find:• ∘ 0• ( −4 )• Define the range of the graph of ∘
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Compositions
Given: = and =Find: ∘ 0
Compositions
Given: = and =Find: ∘ 0
1st: Evaluate = = =
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Compositions
Given: = and =Find: ∘ 0
1st: Evaluate = = =Last: Evaluate = =
∘ =
Compositions
Given: = and =Find: ( −4 )
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Compositions
Given: = and =Find: ( −4 )
1st: Evaluate − = = = −
Compositions
Given: = and =Find: ( −4 )
1st: Evaluate − = = = −Last: Evaluate − = − =− =
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Compositions
Given: = and =Find: Range of ∘
Compositions
Given: = and =Find: Range of ∘
Graph: =
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Compositions
Given: = and =Find: Range of ∘
Graph: = Y = abs(2/(x+2))
Compositions
Given: = and =Find: Range of ∘
Graph: =
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Compositions
Given: = and =Find: Range of ∘
Graph: =: ( , ∞)
or∘ >
Compositions
Given: = + 4 and = − 1Find:• ∘ 9• ( −1 )• Define the domain and range of ( )• Define the domain and range of• Define the domain and range of ∘
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Compositions
Given: = + 4 and = − 1Find: ∘ 9
Compositions
Given: = + 4 and = − 1Find: ∘ 9
1st: Evaluate = + = + =Last: Evaluate = − = − =∘ =
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Compositions
Given: = + 4 and = − 1Find: ( −1 )
Compositions
Given: = + 4 and = − 1Find: ( −1 )
1st: Evaluate − = − − = − =Last: Evaluate = + = + =− =
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Compositions
Given: = + 4 and = − 1Find: Domain and range of ( )
Graph = + 4
Compositions
Given: = + 4 and = − 1Find: Domain and range of ( )
Graph = + 4: [ , ∞) or ≥: [ , ∞) or ( ) ≥
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Compositions
Given: = + 4 and = − 1Find: Domain and range of
Graph = − 1
Compositions
Given: = + 4 and = − 1Find: Domain and range of
Graph = − 1: (−∞, ∞) or X isℝ: [− , ∞) or ≥ −
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Compositions
Given: = + 4 and = − 1Find: Domain and range of ∘
Compositions
Given: = + 4 and = − 1Find: Domain and range of ∘
Graph ∘ = − +In Calc: Y=(x^2 – 1)^(1/2) + 4
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Compositions
Given: = + 4 and = − 1Find: Domain and range of ∘
Graph ∘ = − +: −∞, − ∪ [ , ∞) or ≤ − ≥: [ , ∞) or ∘ ≥
