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1Get sheet of paper2

Solve and graph 3( 3) 7 ( 8)x x

Write answer Interval Notationin your notebooks

: 4: ( 4, )

INQ xINT

Relations and Functions

Chapter 2Section 2-1Pages 72-81

Objectives• I can find Domain and Range• I can determine if the relation is a

function by two methods• IMPORTANT VOCABULARY in

this section!!

Important Vocabulary• Relation• Domain• Range• Function• Vertical Line Test

Cartesian Coordinate Plot

• x-axis: This is the horizontal axis.• y-axis: This is the vertical axis• Origin: This is the center point (0,0)

• Each point on the coordinate plane can be represented by an ordered pair (x,y), where x is the distance from Origin on the X-Axis and y is the distance from Origin on the Y-Axis.

Quadrants

• The coordinate system is divided into 4 quadrants.

• Look at the overhead slide 2-1.1 to see how this entire system works.

1 2 63 4 5 7 8 9 10

432

7

56

89

x-axis

y-axis

0

1-2-6 -3-4-5-7-8-910

-4-3-2-1

-7

-5-6

-8-9

0-1

Quadrant I

(+, +)Quadrant II

(-, +)

Quadrant IV

(+, -)Quadrant III

(-, -)

Origin (0,0)

Relation

• A relation is a set of ordered pairs!

• Need the { } to show a set

• Example: { (1, 2), (3, 4), (5, 6) }

Domain and Range

• The domain in any relation is the first coordinates from the ordered pairs. It is the Input!

• Domain = X -Values• The range in any relation is the second

coordinates from the ordered pairs. It is the Output!

• Range = Y- Values

x-axisDomainInput

Independent Variable

y-axisRange

OutputD

epen

dent

Var

iabl

e

Example 1: Domain/Range

• Given the following relation• {(2,3), (-4,8), (2,6), (7,-3)}• What is the Domain?• { -4, 2, 7}• **Notice they are listed least to greatest!! • No duplicates!!!• What is the Range?• {-3, 3, 6, 8}

Example 2:• Given the following ordered pairs, find

the domain and range.• {(4,5), (-2,3), (5,6)}

• Domain is {-2, 4, 5}• Range is {3, 5, 6}

Answer Format• When listing a set of numbers for domain or range,

use the set symbols {}• List numbers from least to greatest (increasing

order). No duplicates!• Ex: the domain has numbers: 3, -2, 5, 2, 3

• {-2, 2, 3, 5}

SOLUTION

GUIDED PRACTICE for Example 1

1. Identify the domain and range of the function.

Input 0 1 2 4Output 5 2 2 1

The domain is the set of inputs: {0,1,2,4}

The range is the set of outputs: {1,2,5}

4 Ways to see Relations

RelationsOrdered Pairs

(2, 3)

(-3, 1)

(1, -2)

X Y2 3-3 11 -2

Tables

GraphsMapping

2

-3

1

3

1

-2

X Y

Function• A function is a special relation in which• NO DUPLICATED “x-values”• Example: Is the following relation a function:

{ (1,3), (4,-9), (6,3) }• Answer: Yes. Each element from the domain

has only 1 element from the range

Ex 2: How about this relation. Is it a function?

• Given the following { (2,3), (-4,8), (2,6), (7,-3)}• Domain?• { -4, 2, 7}• Range?• {-3, 3, 6, 8}• Function: No. the element 2 from the domain is

mapped to 2 elements in the range.

Tell whether the pairing is a function.

Identify a functionEXAMPLE 2

a.

The pairing is not a function because the input 0 is paired with both 2 and 3.

b.

Identify a functionEXAMPLE 2

OutputInput

21

0 0

4 8

6 12

The pairing is a function because each input is pairedwith exactly one output.

SOLUTION

GUIDED PRACTICE for Example 2

Tell whether the pairing is a function.

1221Output12963Input

2.

36912

1221

The pairing is a function because each input is pairedwith exactly one output.

SOLUTION

GUIDED PRACTICE for Example 2

Tell whether the pairing is a function.

3210Output7422Input

3.

The pairing is not a function because each input is notpaired with exactly one output.

2

47

0123

Vertical Line Test

• You can use a vertical line test to easily see on a graph is the relation is a function.

• You place a straight edge like a pencil vertical on the graph and move it across the graph. If one line intersects the graph at only one point, then it is a function.

Applying VLT

y 2 = x

x

y

Vertical Line Test

Consider the graphs.

x 2 + y

2 = 1

x

y

y = x 2

x

y

2 points of intersection

1 point of intersection

2 points of intersection

Examples VLT

VLT Applies to Grid

½ Sheet Activity

• On your ½ sheet of paper• Make 4 relations that we discussed today.• ONE of each type• Trade with your partner and detrmine

– Domain and Range– Is each a function?

Homework

• WS 1-5