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Chapter 2 Relations and

Functions

Chapter 2.0 Cartesian

Coordinate System

Cross Product

cross product of

Let and be nonempty sets.

The is a

, a

n

nd

d

A B

A B x y x A

B

y

A

B

Cartesian Coordinate System

Consider , and

Each ordered pair of real numbers isassociated with a point in a plane.

R R x y x R y R

Ordered Pairs

Consider an ordered pair ,  which is

associated with point .

- x gives the directed distance of from

the y axis.

- y gives the directed distance of from

the x axis.

x y

P

P

P

Cartesian Coordinate System

axisx

axisy

O

,x yx

y

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Cartesian Coordinate System

c

I

o

f

or

a po

dina

int ,

tes

then and are

the

abscissa

ordinat

of .

:

e:

P x y x y

P

x

y

Example 2.0.1

Plot the following points.

1. 2,7

2. 0, 1

3. 4, 6

4. : with abscissa 3

and ordinate 5

P

Q

R

S

2,7P

0, 1Q

4, 6R

3,5S

Quadrants

1st quadrant2nd Quadrant

3rd Quadrant 4th Quadrant

Distance Formula

1 1 2 2

2 2

2 1 2 1

The distance between two points

, and , is given byP x y Q x y

PQ x x y y

Midpoint

1 1

2 2

1 2 1 2

The midpoint of a line segment

between two points , and

, is

,2 2

P x y

Q x y

x x y y

Slope

1 1 2 2

2 1

2 1

The slope of the line containing

, and , isP x y Q x y

y ym

x x

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Example 2.0.2

2 2

Given 2,7 and 2, 3 ,

1. find the distance between and .

2 2 7 3

16 100

116

4 29

2 29

P Q

P Q

PQ

Given 2,7 and 2, 3 ,

2. find the midpoint of the segment

joining and .

2 2 7 3, 0,2

2 2

P Q

P Q

Given 2,7 and 2, 3 ,

3. determine the slope of the lines

joining and .

3 7 10 5

2 2 4 2

P Q

P Q

m

Chapter 2.1 Relations

Relation

relation from to

Let and be nonempty sets.

A is any

nonempty subset of .

A B

A

A

A B

B

S

S B

Relation in

A relation in is any non-empty

subset of .

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Example

1 2 3 4 5

1 4 9

16 25

Relation

A relation can also be described by

equations and inequalities.

Example

2

2

, is a relation from the set

of nonegative real numbers to .

can also

dependent vari

be described by

is the

is the

able

independent variable

T r A A r

T A r

r

A

Example

1

22

3

21

Following are relations from to .

1. , 2 5

2. ,

3. , 3

4. , 4 1

r x y y x

r x y y x

r x y y x

r x y x y

Graph of a Relation

The is the set

of all points , in a coordinate plane

such that is related to

graph of a re

through

l

t

a

h

t

e

relati

n

io

on .

x y

x y

r

r

Intercepts

is a point where the graph

of a relation crosses the axis.

is a point where the graph

of

i

a

nt

r

erce

elat

pt

in

ion crosses the axis.

tercept

x

y

x

y

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Example

2 2

2

2

Find the and intercepts of

1

intercept: 1,0 , 1,0

if 0, 1

1

intercept: 0,1 , 0, 1

if 0, 1

1

x y

x y

x

y x

x

y

x y

y

Lines

: passing through ,0vertical line

horizontal lin

.

: passing through ,e 0 .

x a a

y a a

Example

Sketch the graph of

1. 3y

3y

2. 2x

2x

Lines

If the defining equation of a relation

is both linear in and , the

linear relatio

relation

is called a and its graph

is a

n

straight l e.in

x y

Example 2.1.8

Identify the and intercepts and

sketch the graph of 2 5.

5intercept: ,0

2

if 0 : 0 2 5

5

2

intercept: 0,5

if 0 : 2 0 5

5

x y

y x

x

y x

x

y

x y

y

2 5y x

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Symmetries

The graph of an equation is symmetric

with respect to the axis if an

equivalent equation is obtained when

is replaced by .

,

,

SWRTY

x y

x y

y

Example 2.1.9

2 2

2 2

2 2

2 2

2 2

Show that the graph of 4

is SWRTY.

4

replacing , by , we get

4

4

Therefore, the graph of 4

is SWRTY.

x y

x y

x y x y

x y

x y

x y

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

,x y ,x y

Symmetries

The graph of an equation is symmetric

with respect to the axis if an

equivalent equation is obtained when

is replaced by .

,

,

SWRTX

x y

x y

x

Example 2.1.10

2

2

2

2

2

Show that the graph of 4

is SWRTX.

4

replacing , by , we get

4

4

Therefore, the graph of 4

is SWRTX.

x y

x y

x y x y

x y

x y

x y

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

,x y

,x y

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Symmetries

The graph of an equation is symmetric

with respect to the origin if an

equivalent equation is obtained when

is replaced by

,

.,

SWRTO

x y

x y

Example 2.1.11

2 2

2 2

2 2

2 2

2 2

Show that the graph of 4

is SWRTO.

4

replacing , by , we get

4

4

Therefore, the graph of 4

is SWRTO.

x y

x y

x y x y

x y

x y

x y

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

,x y

,x y

2

2

2

2

Given , 2 1 ,

1. Find the intercepts.

2 2intercept: ,0 , ,0

2 2

if 0, 0 2 1

2 1

1

2

1 2

22

r x y y x

x

y x

x

x

x

2Given , 2 1 ,

intercept: 0, 1

if 0 : 1

r x y y x

y

x y

2

2

2

2

Given , 2 1 ,

2. Identify the symmetries.

SWRTY

2 1

replacing , by ,

2 1

2 1

The graph is SWRTY.

r x y y x

y x

x y x y

y x

y x

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2

2

2

Given , 2 1 ,

SWRTX

2 1

replacing , by ,

2 1

The graph is not SWRTX.

r x y y x

y x

x y x y

y x

2

2

2

2

Given , 2 1 ,

SWRTO

2 1

replacing , by ,

2 1

2 1

The graph is not SWRTO.

r x y y x

y x

x y x y

y x

y x

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

2Given , 2 1 ,

Symmetry: SWRTY

Intercepts:

2 2: ,0 , ,0

2 2

: 0, 1

r x y y x

x

y

x 1

y 1