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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 ∘ ≥