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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