RelationsA Relation is a rule that links

one or more output to one or more input

An input is anelement in the starter

set. It can also be called x

value

An output is anelement in the arrival

set.It can also be called y

value

Inputs Outputs

FunctionsA relation is a Function if all inputs

are mapped to only one output

This is a function

All inputs are mapped

All inputs have only one output

All inputs are mapped

All inputs have only one output

This is not a function

For example…

State Capital

New Jersey Trenton

New York Albany

Colorado Denver

Rhode Island Providence

Person Grandparent

Eric Arthur

Eric Ellie

Maria Camine

Maria Susan

This tab, where each state ismapped with its capital, represents a function, becauseevery state (inputs) has only onecapital (outputs).

This tab instead, that maps everyperson with his grandparents, isn’t a function, because they both havetwo grandparents (outputs).

x y

5 7

2 9

0 8

-4 6

x y

4 3

2 8

0 8

4 6

Here, each number in the x (inputs) column has only oney (output) as an answer, so itis a function.

Here, the input 4 has anoutput of both 3 and 6, so itcan’t be a function.

How to represent a function

A function can be represented in three ways:1-the graph2-the arrow chart3-the set of ordered pairs

The graph

The graph of a function is a cartesian coordinate system where the couples of x and y values are represented.

this is an exampleof the graph of the function:y=2x+6

The arrow chart

The arrow chart of a function is another graphical representation

The arrows show how the numbers in the domain are mapped into numbers in the range and represent the function f: y=f(x)

The set of ordered pairs

a set of ordered pairs is a non-graphical way to represent a function;the couples of x and y values will be represented as shown in the followingway:

y=2x+6----> {(1,8),(2,10),(3,12)}

X Y

1 3

2 10

3 12

1 Is the following a function?

Yes

No

1

2

3

4

5

4

3

2

A

B

1 Is the following a function?

1

2

3

4

5

4

3

2

Because each input is mapped into only one output

Correct!!

1

1

2

3

4

5

4

3

2

Is the following a function?

Wrong!It is a function because each input is mapped into only one output

2 Is the following a function?

City State

Englewood New Jersey

Englewood Colorado

Springfield Maryland

Springfield Illinois

Springfield Montana

Yes

No

A

B

2 Is the following a function?

City State

Englewood New Jersey

Englewood Colorado

Springfield Maryland

Springfield Illinois

Springfield Montana

Correct!!!

2 Is the following a function?

City State

Englewood New Jersey

Englewood Colorado

Springfield Maryland

Springfield Illinois

Springfield Montana

Wrong!

Vertical Line Test: if every vertical line you can draw goes through only 1 point then the relation is a function.

Function Function Not a function

Example

Not a function

This isn’t a functionbecausethe vertical line goesthroughtwo points.

Is this a function?

Yes

No

A

B

Is this a function?

Correct!!!

Is this a function?

Wrong!

Is this a function?

Yes

No

A

B

Is this a function?

Correct!!!

Is this a function?

Wrong!

Is this a function?

Yes

No

A

B

Is this a function?

Correct!!!

Is this a function?

Wrong!

Domain codomain and range

Any set of numbers that is the input of a relation or function is called the domain.

Any set of numbers that is the output of a relation or function is called the range.

Domain: {1, 2, 3, 4} Range: {42, 98, 106, 125}

f(x) Y=2x+1

32

DOMAIN: a set of first elementin a relation ( all of the x values ). These are also called the independent variable.

RANGE:the second and actualselements in a relation ( all of the y values ). These are also calledthe dependent variable

1

3

4

5

7

9

CODOMAIN: is the set of all possible and second elements in a relation.

• Domain = {Joe, Mike, Rose, Kiki}

• Range = {6, 5.75, 5, 6.5}

Example

Joe

Mike

Rose

kiki

DOMAIN RANGE

6

5,75

5

6,75

Domain (set of all x’s) Range (set of all y’s)

1

2

3

4

5

2

10

8

6

4

Which numbers are in the domain?

2

4

6

8

10

10

20

30

40

50

2

10

12

8

20

A

B

C

D

E

Which numbers are in the domain?

2

4

6

8

10

10

20

30

40

50

2

10

8

A

B

D

Which numbers are in the range?

2

4

6

8

10

10

20

30

40

50

2

10

12

8

20

A

B

C

D

E

Which numbers are in the range?

2

4

6

8

10

10

20

30

40

50

10

20

B

E

Which numbers are part of the domain?

{(24, 12), (22, 11), (20, 10), (18, 9)}

12

20

24

11

9

A

B

C

D

E

Which numbers are part of the domain?

{(24, 12), (22, 11), (20, 10), (18, 9)}

20

24

B

C

Which numbers are part of the range?

{(24, 12), (22, 11), (20, 10), (18, 9)}

12

20

24

11

9

A

B

C

D

E

Which numbers are part of the range?

{(24, 12), (22, 11), (20, 10), (18, 9)}

12

11

9

A

D

E

INJECTIVE, SURJECTIVE and BIJECTIVE FUNCTIONS.

•

•

•

• •

•

• •

X

Y

Let f be a function whose domain is a set X. The function f is injective if for all a and b in X, if f(a) = f(b) then a = b; Equivalently, if a ≠ b, then f(a) ≠ f(b).So….F(x) is injective (also called one-to-one) if no twoinputs have the same output.

This is an INJECTIVE function.

This isn’t an INJECTIVE function because we can

see that two differentelements ‘3;4’ in the

domain have the sameoutput ‘C’.

To be a one-to-one function, each y value could only be

paired with one x. Let’s look at a couple of graphs.

Look at a y value (for example

y = 3) and see if there is only

one x value on the graph for it.

This is NOT a one-to-one

function

For any y value, a horizontal line

will only intersect the graph once

so will only have one x value

This then IS a one-to-one

function

If a horizontal line intersects the graph of an

equation more than one time, the equation

graphed is NOT a one-to-one function.

This is an injective

funtion.

This is NOT an

injective funtion.

This is NOT an

injective funtion.

•

•

• •

•

•

•

X

Y

A function f (from set X to Y) is surjective if and only if forevery b in Y, there is at least one a in X.So…A function is surjective (onto) if every element in the codomain (Y) has some element from the domain (X) mappedon to it.In other words: f(x) is surjective when the range is equal to the codomain.

It’s surjective because allelements in the codomain

has an input

It isn’t surjective becausethere is an element in the codomain Y that hasn’t an

input

•

•

•

•

•

•

A

B

A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y .So…A function is bijective if it’s both injective and surjective.

•Is injective because each input has a different output.

•is surjective because the range is allcodomain.

It’s BIJECTIVE

a

b

c

1

2

This function is:•injective•surjective•bijective•neither

a

b

c

1

2

3

This function is:•injective•surjective•bijective•neither

Davide

Lia

Ugo

Irene

Marco

Erika

This function is:•injective•surjective•bijective•neither

Tom

Gigi

Carla

Sammy

Anna

This function is:•injective•surjective•bijective•neither

A: {Maria; Francesca; Vanessa; Gaia}B: {Edoardo; Andrea; Luca; Davide}

A B

Maria Andrea

Vanessa Edoardo

Francesca Luca

Gaia Davide

This is a function:•Injective•Surjective•Bijective

A: {Maria; Vanessa; Gaia}B: {Edoardo; Andrea; Luca; Davide}

A B

Maria Andrea

Vanessa Edoardo

Gaia Davide

This is a function:•Injective•Surjective•Bijective

A: {Maria; Francesca; Vanessa; Gaia; Ilaria}B: {Edoardo; Andrea; Luca; Davide}

A B

Maria Andrea

Vanessa Edoardo

Francesca Luca

Gaia Edoardo

Ilaria Davide

This is a function:•Injective•Surjective•Bijective