Aim: What are the relation and function?

Do Now: The following table describes the tests

scores and the names of studentsscores Names

99 Al

96 Bob

98 Cathy

97 Debbie

95 Eric

If the first ordered pair is (99, Al) write the rest ordered pairs

HW:

p.126 # 3,4,5,6,8,10

p.134 # 8,10,12,14,18

The set of ordered pairs is called relation

The domain of a relation is the set consisting of all first elements of the ordered pairs

In the Do Now, the domain is the set of scores

{99, 96, 98, 97, 95}

The range of a relation is the set consisting of all second elements of the ordered pairs

In the Do Now, the range is the set of names

{Al, Bob, Cathy, Debbie, Eric}

Another example, the following is a relation:Another example, the following is a relation:

)7,6( ),4,2( ),3,2( ),2,1(

-3

-2

-1

0

6

7

8

9

domain range1

2

3

4

-5

5

6

domain range

Example:

State the domain and range of the following relation.

{(2, –3), (4, 6), (3, –1), (6, 6), (7, 3)}

domain:  {2, 3, 4, 6,7}range:  {–3, –1, 3, 6}

State the domain and range of the following relation.

{(5, –3), (5, 6), (8, –1), (2, 6), (7, 9)}

Domain: {5,8,2,7}

Range: {-3,6,-1,9}

There are two types of relation: finite and infinite

Example for finite relation:

1

2

3

a

b

c

Domain Range

Examples for Infinite relation:

relation A = {(x,y)| y = x + 1} relation B= {(x,y)|x < y}

In both relations, there are infinitely many ordered pairs.

There are only three ordered pairs.

2

4

5

12345

6

The above diagram describes a certain relation

a. Write the indicate relation as a set of ordered pairs

b. State the domain of the relation

c. State the range of the relation

Example:

graph the equations by table of values

x y

0

1

-1

2

-2

x y

0

1

2

-1

-2

y = x y = x2

We can write the relations for both equations based on the tables

y = x

)}2,2)(2,2)(1,1)(1,1)(0,0{(

y = x2

)}4,2)(1,1)(4,2)(1,1)(0,0{(

For both relations, each element in the domain corresponds only one element in the range.

Therefore we say line and parabola are both functions.

If a relation is a function, If a relation is a function, thenthen each value of x can only have one corresponding y, but y can have more than one value of x.

For y = x, each x corresponds exactly one y, we call this type of function as one to one function.

On the contrary, y = x2, some values of x correspond to the same value of y, then this is NOT one to one function

A Function is a relation in which no two ordered pairs have the same first element.

{(2,3)(3,4)(1,-2)(5,6)(7,8)}

This relation is a function since each x only corresponds one y

{(2,4)(9,1)(7,-2)(5,6)(-4,1)}This relation is a function since x correspond only one value of y although y corresponds two different x

{(1,2)(1,3)(4,6)(7,-5)(10,11)}

This relation is NOT a function since the same value of x corresponds different y’s.

Math Composer 1. 1. 5http: / / www. mathcomposer. com

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y = x

If we draw a vertical line anywhere on the coordinate plane, the vertical line only intersect the graph of y = x at only one point. This is called Vertical line test

Horizontal line test: If we draw a horizontal line anywhere on the coordinate plane, the line only intersect the graph at one point, then we call y = x is a one-to-one function

Math Composer 1. 1. 5http: / / www. mathcomposer. com

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If we draw a vertical line anywhere on the coordinate plane, the vertical line only intersects the graph of y = x2 at only one point

We can use the diagram to show if the relation is a function

1

222

3

a

b

c

One to one function

1

2

3

a

b

c

A function (not 1 to 1)

1

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a

b

c

Not a Function

Drill:

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In the following questions, state whether or not each indicated relation is a function

1..1

2

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7

2.

1

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6

3.

1

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4.

Drill:Which relation, if either, is a function?

A = {(0,3), (1,8), (1,5)} B = {(0,2), (1,2), (3,2)}

C = {(1,3), (2,3), (4,3), (5,4)}

D = {(2,4), (1,4), (1,6), (3,5)}

E= {(2,5), (4,1), (3,1), (6,1), (7,5)}

Drill: Which graph represents a relation that is a function?Math Composer 1. 1. 5http: / / www. mathcomposer. com

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Math Composer 1. 1. 5http: / / www. mathcomposer. com

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Math Composer 1. 1. 5http: / /www. mathcomposer. com

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