Section 1.6 Inverses

What Is A Function?

Who can tell me what is a function?

DefinitionFor nonempty sets A and B, a function f maps A to B denotedf : A→ B, such that for each element a ∈ A, there is exactly oneelement b ∈ B. We write this as f (a) = b of b is the unique element ofB that is assigned to a ∈ A.

What functions can you name?

What Is A Function?

Who can tell me what is a function?

DefinitionFor nonempty sets A and B, a function f maps A to B denotedf : A→ B, such that for each element a ∈ A, there is exactly oneelement b ∈ B. We write this as f (a) = b of b is the unique element ofB that is assigned to a ∈ A.

What functions can you name?

What Is A Function?

Who can tell me what is a function?

DefinitionFor nonempty sets A and B, a function f maps A to B denotedf : A→ B, such that for each element a ∈ A, there is exactly oneelement b ∈ B. We write this as f (a) = b of b is the unique element ofB that is assigned to a ∈ A.

What functions can you name?

Important Terms

Let f : X → Y be a function mapping X to Y .

DefinitionX is the domain of f .

Important Terms

Let f : X → Y be a function mapping X to Y .

DefinitionX is the domain of f .

Important Terms

What is the Y? (blue region here)

DefinitionY is the codomain of f .

Important Terms

What is the Y? (blue region here)

DefinitionY is the codomain of f .

Important Terms

So what is the yellow region here?

DefinitionThe range of f is the set of all f (x) where x ∈ X.

Important Terms

So what is the yellow region here?

DefinitionThe range of f is the set of all f (x) where x ∈ X.

In Terms of the Elements

Anyone know what we call x and y here?

Definitiony = f (x) is called the image of x.

So, the range is the set of all images of X.

In Terms of the Elements

Anyone know what we call x and y here?

Definitiony = f (x) is called the image of x.

So, the range is the set of all images of X.

In Terms of the Elements

Anyone know what we call x and y here?

Definitiony = f (x) is called the image of x.

So, the range is the set of all images of X.

In Terms of the Elements

Definitionx is called the preimage of y = f (x).

Why We Bring This Up

A function can only have an inverse if it is 1-1. Do we rememberwhat 1-1 means?

Inverses

Definition

If f (a) = b, then the function f−1 is the inverse function of f andf−1(b) = a.

So what we are saying is that we need to not only have a uniqueoutput for each input (function) but we need to also have a uniqueinput for each output.

Inverses

Definition

If f (a) = b, then the function f−1 is the inverse function of f andf−1(b) = a.

So what we are saying is that we need to not only have a uniqueoutput for each input (function) but we need to also have a uniqueinput for each output.

Inverses Visually

x

y

Notice that the domain of the original function equals the range of theinverse and the domain of the inverse equals the range of the originalfunction ...

Inverses Visually

x

y

Notice that the domain of the original function equals the range of theinverse and the domain of the inverse equals the range of the originalfunction ...

Inverses from Tables

ExampleGiven the following table,

x 2 4 6 8f (x) 8 6 2 4

Find f (6) and f−1(4)

f (6) = 2

f−1(4) = 8

Inverses from Tables

ExampleGiven the following table,

x 2 4 6 8f (x) 8 6 2 4

Find f (6) and f−1(4)

f (6) =

2

f−1(4) = 8

Inverses from Tables

ExampleGiven the following table,

x 2 4 6 8f (x) 8 6 2 4

Find f (6) and f−1(4)

f (6) = 2

f−1(4) = 8

Inverses from Tables

ExampleGiven the following table,

x 2 4 6 8f (x) 8 6 2 4

Find f (6) and f−1(4)

f (6) = 2

f−1(4) =

8

Inverses from Tables

ExampleGiven the following table,

x 2 4 6 8f (x) 8 6 2 4

Find f (6) and f−1(4)

f (6) = 2

f−1(4) = 8

Inverses from Graphs

ExampleFor the given graph,

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-2

-1

f (x)

find f (2) and f−1(4)

f (2) = 1f−1(4) = -1 and 3

Inverses from Graphs

ExampleFor the given graph,

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-2

-1

f (x)

find f (2) and f−1(4)

f (2) =

1f−1(4) = -1 and 3

Inverses from Graphs

ExampleFor the given graph,

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-2

-1

f (x)

find f (2) and f−1(4)

f (2) = 1

f−1(4) = -1 and 3

Inverses from Graphs

ExampleFor the given graph,

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-2

-1

f (x)

find f (2) and f−1(4)

f (2) = 1f−1(4) =

-1 and 3

Inverses from Graphs

ExampleFor the given graph,

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-2

-1

f (x)

find f (2) and f−1(4)

f (2) = 1f−1(4) = -1 and 3

Finding Inverses

This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.

Example

Find the inverse of f (x) = 3x + 2.

1 Rewrite as ‘y =’y = 3x + 2

2 Switch x and yx = 3y + 2

3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2

3 = y4 Rewrite as ‘f−1(x) =’

f−1(x) = x−23

Finding Inverses

This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.

Example

Find the inverse of f (x) = 3x + 2.

1 Rewrite as ‘y =’y = 3x + 2

2 Switch x and yx = 3y + 2

3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2

3 = y4 Rewrite as ‘f−1(x) =’

f−1(x) = x−23

Finding Inverses

This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.

Example

Find the inverse of f (x) = 3x + 2.

1 Rewrite as ‘y =’y = 3x + 2

2 Switch x and yx = 3y + 2

3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2

3 = y4 Rewrite as ‘f−1(x) =’

f−1(x) = x−23

Finding Inverses

This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.

Example

Find the inverse of f (x) = 3x + 2.

1 Rewrite as ‘y =’y = 3x + 2

2 Switch x and yx = 3y + 2

3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2

3 = y4 Rewrite as ‘f−1(x) =’

f−1(x) = x−23

Finding Inverses

This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.

Example

Find the inverse of f (x) = 3x + 2.

1 Rewrite as ‘y =’y = 3x + 2

2 Switch x and yx = 3y + 2

3 Solve for y

x = 3y + 2⇒ x− 2 = 3y⇒ x−23 = y

4 Rewrite as ‘f−1(x) =’f−1(x) = x−2

3

Finding Inverses

This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.

Example

Find the inverse of f (x) = 3x + 2.

1 Rewrite as ‘y =’y = 3x + 2

2 Switch x and yx = 3y + 2

3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2

3 = y

4 Rewrite as ‘f−1(x) =’f−1(x) = x−2

3

Finding Inverses

This process will always find the inverse relation - question will bewhether or not we need to restrict the domains.

Example

Find the inverse of f (x) = 3x + 2.

1 Rewrite as ‘y =’y = 3x + 2

2 Switch x and yx = 3y + 2

3 Solve for yx = 3y + 2⇒ x− 2 = 3y⇒ x−2

3 = y4 Rewrite as ‘f−1(x) =’

f−1(x) = x−23

Plot of f and f−1

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-5

-4

-3

-2

-1

f (x)

f−1(x)

Domain of f−1(x)? R

Plot of f and f−1

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-5

-4

-3

-2

-1

f (x)

f−1(x)

Domain of f−1(x)? R

Plot of f and f−1

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-5

-4

-3

-2

-1

f (x)

f−1(x)

Domain of f−1(x)? R

Plot of f and f−1

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-5

-4

-3

-2

-1

f (x)

f−1(x)

Domain of f−1(x)?

R

Plot of f and f−1

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-5

-4

-3

-2

-1

f (x)

f−1(x)

Domain of f−1(x)? R

Finding Inverses

Example

Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.

How do we know this f (x) is not invertible without restricting thedomain?

It is not 1-1 ...

y = x2 − 3

x = y2 − 3

x + 3 = y2

y = ±√

x + 3

Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =

√x + 3 and D(f−1) = [−3,∞).

Finding Inverses

Example

Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.

How do we know this f (x) is not invertible without restricting thedomain?

It is not 1-1 ...

y = x2 − 3

x = y2 − 3

x + 3 = y2

y = ±√

x + 3

Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =

√x + 3 and D(f−1) = [−3,∞).

Finding Inverses

Example

Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.

How do we know this f (x) is not invertible without restricting thedomain?

It is not 1-1 ...

y = x2 − 3

x = y2 − 3

x + 3 = y2

y = ±√

x + 3

Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =

√x + 3 and D(f−1) = [−3,∞).

Finding Inverses

Example

Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.

How do we know this f (x) is not invertible without restricting thedomain?

It is not 1-1 ...

y = x2 − 3

x = y2 − 3

x + 3 = y2

y = ±√

x + 3

Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =

√x + 3 and D(f−1) = [−3,∞).

Finding Inverses

Example

Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.

How do we know this f (x) is not invertible without restricting thedomain?

It is not 1-1 ...

y = x2 − 3

x = y2 − 3

x + 3 = y2

y = ±√

x + 3

Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =

√x + 3 and D(f−1) = [−3,∞).

Finding Inverses

Example

Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.

How do we know this f (x) is not invertible without restricting thedomain?

It is not 1-1 ...

y = x2 − 3

x = y2 − 3

x + 3 = y2

y = ±√

x + 3

Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =

√x + 3 and D(f−1) = [−3,∞).

Finding Inverses

Example

Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.

How do we know this f (x) is not invertible without restricting thedomain?

It is not 1-1 ...

y = x2 − 3

x = y2 − 3

x + 3 = y2

y = ±√

x + 3

Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =

√x + 3 and D(f−1) = [−3,∞).

Finding Inverses

Example

Find the inverse of f (x) = x2 − 3 on [0,∞) and find the appropriatedomain.

How do we know this f (x) is not invertible without restricting thedomain?

It is not 1-1 ...

y = x2 − 3

x = y2 − 3

x + 3 = y2

y = ±√

x + 3

Since we have nonnegative x, we have that the inverse we need is thepositive root. So, f−1(x) =

√x + 3 and D(f−1) = [−3,∞).

Graph of f and f−1

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-5

-4

-3

-2

-1

f (x)

f−1(x)

Graph of f and f−1

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-5

-4

-3

-2

-1

f (x)

f−1(x)

Graph of f and f−1

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

4

3

2

1

-5

-4

-3

-2

-1

f (x)

f−1(x)