Math 71B 9.2 – Composite and Inverse Functions 1.
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Transcript of Math 71B 9.2 – Composite and Inverse Functions 1.
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Math 71B
9.2 – Composite and Inverse Functions
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When you plug a function into another function, it’s called _____________________________.
ex: If and , then _____________________________𝒇 (𝟑 𝒙 −𝟐 )= (𝟑𝒙 −𝟐 )𝟐
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When you plug a function into another function, it’s called _____________________________.
ex: If and , then _____________________________
function composition
𝒇 (𝟑 𝒙 −𝟐 )= (𝟑𝒙 −𝟐 )𝟐
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When you plug a function into another function, it’s called _____________________________.
ex: If and , then _____________________________
function composition
𝒇 (𝟑 𝒙 −𝟐 )= (𝟑𝒙 −𝟐 )𝟐
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When you plug a function into another function, it’s called _____________________________.
ex: If and , then _____________________________
function composition
𝒇 (𝟑 𝒙 −𝟐 )= (𝟑𝒙 −𝟐 )𝟐
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Notation: ____________
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Notation: ____________ 𝒇 (𝒈 (𝒙 ) )
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The domain of is the set of all ___ in the domain of ___ such that _____ is in the domain of ___.
𝑔 𝑓
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The domain of is the set of all ___ in the domain of ___ such that _____ is in the domain of ___.
𝑔 𝑓
𝒙
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The domain of is the set of all ___ in the domain of ___ such that _____ is in the domain of ___.
𝑔 𝑓
𝒙𝒈
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The domain of is the set of all ___ in the domain of ___ such that _____ is in the domain of ___.
𝑔 𝑓
𝒙𝒈 𝒈 (𝒙 )
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The domain of is the set of all ___ in the domain of ___ such that _____ is in the domain of ___.
𝑔 𝑓
𝒙𝒈 𝒈 (𝒙 ) 𝒇
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The domain of is the set of all ___ in the domain of ___ such that _____ is in the domain of ___.
𝑔 𝑓
𝒙𝒈 𝒈 (𝒙 ) 𝒇
𝒙
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The domain of is the set of all ___ in the domain of ___ such that _____ is in the domain of ___.
𝑔 𝑓
𝒙𝒈 𝒈 (𝒙 ) 𝒇
𝒙 𝒈 (𝒙 )
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The domain of is the set of all ___ in the domain of ___ such that _____ is in the domain of ___.
𝑔 𝑓
𝒙𝒈 𝒈 (𝒙 ) 𝒇
𝒙 𝒈 (𝒙 ) 𝒇 (𝒈 (𝒙 ))
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Ex 1.Let and . Find and .
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Some function “undo” each other. Like and .
Functions like this are called _____________.
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Some function “undo” each other. Like and .
Functions like this are called _____________.inverses
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What happens when we compose inverses?
Let’s try with and :
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What happens when we compose inverses?
Let’s try with and :
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What happens when we compose inverses?
Let’s try with and :
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What happens when we compose inverses?
Let’s try with and :
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What happens when we compose inverses?
Let’s try with and :
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Here’s the formal definition: and are inverse functions if both
1. ____ (for every in the domain of )
2. ____ (for every in the domain of )
Notation: The inverse of is written .
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Here’s the formal definition: and are inverse functions if both
1. ____ (for every in the domain of )
2. ____ (for every in the domain of )
Notation: The inverse of is written .
𝒙
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Here’s the formal definition: and are inverse functions if both
1. ____ (for every in the domain of )
2. ____ (for every in the domain of )
Notation: The inverse of is written .
𝒙
𝒙
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Here’s the formal definition: and are inverse functions if both
1. ____ (for every in the domain of )
2. ____ (for every in the domain of )
Notation: The inverse of is written .
𝒙
𝒙
Does not mean
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Ex 2.Show that and are inverses of each other.
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How to find the inverse of :
1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________.
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How to find the inverse of :
1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________.
𝒚
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How to find the inverse of :
1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________.
𝒚Switch
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How to find the inverse of :
1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________.
𝒚Switch
𝒚
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How to find the inverse of :
1. Replace with ___.2. __________ and .3. Solve for ____.4. Replace with _________.
𝒚Switch
𝒚𝒇 −𝟏(𝒙)
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Ex 3.Find the inverse of
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Given any point on the graph of , we can get a point on the graph of by switching the coordinates: .
Graphs and Inverses
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Ex 4.Given the graph of , draw the graph of .
Graphs and Inverses
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Ex 4.Given the graph of , draw the graph of .
Graphs and Inverses
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Ex 4.Given the graph of , draw the graph of .
Graphs and Inverses
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Ex 4.Given the graph of , draw the graph of .
Graphs and Inverses
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Ex 4.Given the graph of , draw the graph of .
Graphs and Inverses
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Notice that the entire graph of will be the mirror image of across the line .
Graphs and Inverses
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The _____________________________ is a visual way to determine if a function has an inverse.
Ex 5.Do the following graphs have inverses?
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The _____________________________ is a visual way to determine if a function has an inverse.
Ex 5.Do the following graphs have inverses?
horizontal line test
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The _____________________________ is a visual way to determine if a function has an inverse.
Ex 5.Do the following graphs have inverses?
horizontal line test
Yes
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The _____________________________ is a visual way to determine if a function has an inverse.
Ex 5.Do the following graphs have inverses?
horizontal line test
Yes No
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The _____________________________ is a visual way to determine if a function has an inverse.
Ex 5.Do the following graphs have inverses?
horizontal line test
Yes No
No
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The _____________________________ is a visual way to determine if a function has an inverse.
Ex 5.Do the following graphs have inverses?
horizontal line test
Yes
Yes
No
No
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Note: Functions that pass the horizontal line test are called ____________ functions.
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Note: Functions that pass the horizontal line test are called ____________ functions.one-to-one