Chapter 2 Relations and

Functions

Objectives

1. define relations, functions and inverse functions;

2. state the domain, range, intercepts and symmetry of the functions and relations;

3. differentiate relations from functions;

4. perform operations on functions; and

5. sketch the graphs of functions and their inverses.

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

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

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 .

Example 2.1.1

1,1 , 2,4 , 3,9 , 4,16 , 5,25is a relation from to .

S

1 2 3 4 5

1 4 9

16 25

Relation

A relation can also be described by

equations and inequalities.

Example 2.1.2

2

2

, is a relation from the set

of nonegative real numbers to .

can also

independent va

be describ

riable

ed by

is the

i deps the endent variable

T r A A r

T A r

r

A

Example 2.1.3

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

Domain

The of a relation : , denoted

by , is the set containing all the first

members of the or

dom

dered pai s

a

r .

in r A B

Dom r

Range

The of , denoted by is

the set containing all the second

members of the ordered pa

r

i

a

rs i

e

n .

ng r Rng r

r

Example 2.1.4

Identify the domain and range of the

relation

1,1 , 2,4 , 3,9 , 4,16 , 5,25

1,2,3,4,5

1,4,9,16,25

S

Dom S

Rng S

Domain and Range

The domain is the set of all values of the independent variable

permisible

resulting

.

The range is the set of all values of the dependent variable.

Example 2.1.5

Identify the domain and range of the

following relations.

1. , 2 1S x y y x

Dom S

Rng S

2

2

2. ,

0,

3. , 4

4,

T x y y x

Dom T

Rng T

U x y y x

Dom U

Rng U

24. ,

1

1

0

5. , 1

1 0 1,

1 0,

V x y yx

Dom V

Rng V

W x y y x

x Dom W

x Rng W

26. ,

0,

0

7. , 2 3

0,

X x y x y

y x Dom X

x Rng X

Y x y y x

Dom Y

Rng Y

8. , 5 4

4,

9. , 5

0

0,

,5

Z x y y x

Dom Z

Rng Z

A x y y x

x

Dom A

Rng A

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

Example 2.1.6

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 2.1.7

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

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

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

Example 2.1.11

2

2

2

2

Given , 2 1 ,

1. Find the domain and range.

1,

0

2 0

2 1 1

r x y y x

Dom r Rng r

x

x

x

2

2

2

2

Given , 2 1 ,

2. 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 ,

3. 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

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 ,

1,

Symmetry: SWRTY

Intercepts:

2 2: ,0 , ,0

2 2

: 0, 1

r x y y x

Dom r

Rng r

x

y

x 1

y 1

Special Graphs

2

2

Parabola

: parabola opening

upward if 0

downward if 0

: parabola opening to the

left if 0

right if 0

y ax k

a

a

x ay k

a

a

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Example 2.1.12

2Given 1, find the domain and range

then sketch the graph.

Domain: 1,

Range:

x y

x 2 1 2

y -1 0 1

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Example 2.1.13

2Given 2, find the domain and range

then sketch the graph.

Domain:

Range: 2,

y x

X -1 0 1

y -1 -2 -1

Special Graphs

2 2 2

centered

Circle

at 0,0 radi

s

, 0

Circle wi u th .s

x y a

a

a

a

a

a

a

Domain: ,

Range: ,

a a

a a

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Example 2.1.14

2 2Given 16, find the domain and range

then sketch the graph.

Domain: 4,4

Range: 4,4

x y

Special Graphs

2 2

2 2

Ellipse

1

intercepts: ,0 , ,0

intercepts: 0, , 0,

Domain: ,

Range: ,

x y

a b

x a a

y b b

a a

b b

aa

b

b

-4 -2 2 4

-4

-2

2

4

Example 2.1.15

2 2

2 2

2 2

Given 4 9 36, find the domain and range

then sketch the graph.

4 9 36

19 4

intercepts: 3,0 , 3,0

intercepts: 0,2 , 0, 2

Domain: 3,3

Range: 2,2

x y

x y

x y

x

y

Special Graphs

2 2

2 2

2 2 2 22

2 2 2

2 2

2 2

Hyperbola

1 intercepts: ,0 , ,0

Asymptotes:

0

x yx a a

a b

x y b xy

a b a

y x by x

b a a

Special Graphs

Hyperbola

intercepts: ,0 , ,0

Asymptotes:

Domain: , ,

Range:

x a a

by x

a

a a

aa

b

b

by x

a

by x

a

Example 2.1.16

2 2

2

2

Given 1, find the domain and range9 25

then sketch the graph.

intercept: 3,0 , 3,0

if 0, 19

9

3

x y

x

xy

x

x

2 2

2

2

19 25

intercept: 3,0 , 3,0

intercept: none

if 0, 125

25

5Asymptotes:

3

x y

x

y

yx

y

y x

-4 -2 2 4

-4

-2

2

4

5

3y x

5

3y x

Domain: , 3 3,Range: