Do NowEvaluate the function: 3 4g x x

1g h x

2 4h x x x

33 h x

HomeworkNeed help?

Look in section 7.7 – Inverse Relations & Functions

in your textbook

Worksheet: Inverses WS

Unit 3: Domain & RangeDay 12: Inverses

ObjectiveTo find the inverse of a relation using:

A mapping diagramA tableAn equation

To determine if an inverse relation is a function using:A mapping diagramA tableA graphAn equation

Inverse RelationsWhat is an inverse relation?

A relation that “undoes” a relation or function

The inverse of any relation is written as:

Domain (x –

values)Range

(y – values)

2 4y x x f(x)

-10

1

2

2

4

6

8

f(x) x

2

4

6

8

-10

1

2

4

2

yx

Function:

Inverse:

1f x“inverse of

f”

Inverse RelationsAnother way to think about it:

Inverses switch the domain & range of a relation

DOMAIN

RANGE-1012

2468

Relation: (-1, 2), (0, 4), (1, 6), (2, 8) Inverse: (2, -1), (4, 0), (6, 1), (8, 2) Relation

DOMAIN

RANGE

-1012

2468

Inverse

Is the relation and/or the inverse a function?

Yes, all elements in both domains correspond to one element in the range

Find the inverse of the given relation.State if the relation and/or its inverse are functions.

DOMAIN

RANGE

4 05-3

2-69

x y

3 5

-6 -3

2 1

-6 -4

Inverse: Inverse:

x y

5 3

-3 -6

1 2

-4 -6

DOMAIN

RANGE

4 05-3

2-69

FUNCTION

NOT A FUNCTION

NOT A FUNCTION

FUNCTION

Step 3: solve for yStep 2: switch x and

y

Find the inverse of the function.

Step 1: put “y” for “f(x)”

4 8f x x 4 8y x 4 8x y 8 4x y

4

4y

8

4

x 1

24x y

1 12

4f x x Step 4: Write using

inverse notation

Determine if the inverse is a function.

Yes, the inverse is a function because all lines pass the VLT.

Inverse Relations as FunctionsIf two equations are inverses, you will be able

to apply the function, then its inverse and get back to the original value.

Find

Find

2 4f x x 1 4

2

xf x

1 6f f

6 2 6 4f 16 1 16 416

2f

6

1 2f f

3 2 3 4f 1 2 42

2f

23

Both f(g(x)) = x and g(f(x)) = x to be inverses.

Find the inverse and determine if it is a function.

Determine if the two functions are inverses.

14 2f x x

14 2y x 14 2x y

41 4

1 2x

4 8x y

1 4 8f x x

142x y

12

58

1x

g x x

f x

124 4 1g

1

1 51

8f

6

8

Choose any value for x and see if f g x x

3

4

No, the functions are not inverses.

Yes, the inverse is a function because all lines are functions.

Both f(g(x)) = x and g(f(x)) = x to be inverses.

Inverse Relations 2 4f x x

1 4

2

xf x

Function:

Inverse:

x f(x)

-1 2

0 4

1 6

2 8

x f -1(x)

2 -1

4 0

6 1

8 2

All inverses have a line of symmetry at y

= x

Determine if the graphs depict functions and their inverses.

Yes, the graphs show functions & inverses because there is a line of symmetry at y = x.

No, the graphs do not show functions & inverses because there is no line of symmetry at y = x.

Properties of InversesIf a relation maps a onto b; then an inverse

“undoes” the relation and maps b back onto a

The domain of the relation is the range of the inverse.

The range of the relation is the domain of the inverse.

Graphs of relations & inverses have a line of symmetry at the equation y = x

Domain of fRange of f

-1

Range of f

Domain of f -1

a b

f

f -1

1f f xx x

1f xf x x

Did you meet today’s objective?What does an inverse do?Describe how to find an inverse when given:

A tableA mapping diagramA graphAn equation