1.1 (relation,function and linear function)

8/3/2019 1.1 (Relation,Function and Linear Function)

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CHAPTER 1



Relations andFunctions



Cross Product

Let A and B be sets.

The cross product of A and B, denoted by

A x B is the set of all ordered pairs (x, y)such that x A and y B.

Notation:

A x B = {(x, y)` x A and y B}



Examples:

1.

2.



Relations

A relation from A to B is anynonempty subset of A x B .



1. Any set of points in the rectangularCartesian coordinates is a relation.

Examples:

A = {(1,2),(3,5),(4,6),(8,9)}

B = {(1,0),(2,0),(1,5)}



2.Equation and inequality are relations.

3. Tables, graphs and mappings also

represent relations.



FunctionsFunctions

Definition:

A function f from A to B is a relation from A to B

where to each a A, there corresponds exactly

one b B.

Alternative Definition:A function is a set of ordered pairs in which no two

ordered pairs have the same first component.



1.

Examples: Identify which of the following

are functions.

A = {(1,2),(3,5),(4,5),(8,9)}

B = {(1,0),(2,0),(1,5)}

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

D = {(-1,2),(1,3),(2,3)}



2.



V

ertical Line TestGiven the graph of a relation, if

any vertical line constructedintersects (or passes) the graphin at most one point, then the

relation described by the graphis that of a function.



Domain and Range

Consider the function

The set of all x such that isthe domain of f, denoted by

The set of all y such that isthe range of f, denoted by



The domain is also defined as the set of

all permissible values of x andrange as the set of all correspondingvalues of y.

Domain and Range



Examples:

1. A = {(1,2),(3,5),(4,5),(8,9)}

Domain: {1, 3, 4, 8} Range:{2, 5, 9}

2.



Functions

Two main types of functions:

Algebraic functions ² those functions that can

be obtained by a finite combination of constants

and variables together with the four basic

operations, exponentiation, or root extractions.

Transcendental functions ² those that are not

algebraic.



AlgebraicFunctions



POLYNOMIAL FUNCTIONSGeneral Form:

where n is a non-negative integer and

012

23

32

21

1 a xa xa xa xa xa xa x P nn

nn

nn !

-

a0,a1,a2,..., an R

The domain of any polynomial function is the

set of all real numbers.

If , the polynomial function P is said

to be of degree n.

0{na



Standard Form:

, where b is a real number.

Constant Functions

y f x b! !

Domain:Range:

f D !

_ a f R b!

Graph:Horizontal line passing through (0 ,b).



Constant Functions

Examples

Find the domain, range and sketch the

Graph of the following:

1. 2 y f x! !

2. 3 y g x! !



Linear Functions

Standard Form:

where m and b are any real numbers.

Domain:

Range:

or

or

Graph: Straight line



Examples of Linear Function

4

523

!

x ) x( h.

2. ( ) 3 g x x!

421 ! x ) x( f .

Non-examples

42 . ( )

2 5i x

x!

11. ( )n x

x!



How do we sketch the graph of a

linear function?

It·s enough to sketch a line using any 2

points.

We can use

y-intercept: the value of y when x=0 x-intercept: the value of x when y=0



Note:

y-intercept: the value of y when x=0

x-intercept: the value of x when y=0

Thus , if , then

y-intercept: -4

x-intercept: 2

( ) 2 4 y g x x! !

y-intercept: y=-4

x-intercept: x=2



Illustrations:421 ! x ) x( f .

y-intercept: -4

x-intercept: 2

4

,0

0,2



Illustrations:y-intercept: 4

x-intercept: 4 x ) x( g . ! 42

4,0

0,4



Illustrations:y-intercept: ?

x-intercept: ?

¹ º ¸©

ª¨

4

50 ,

¹ º ¸©

ª¨ 0

2

5 ,

4

523

!

x ) x( h.

int :

5 50, int :4 4

int :

0, 0 2 5

2 5

5 5int :2 2

f or y

when x y y

f or x

when y x

x

x x

! !

! !

!

!



Illustrations:

4. ( ) g x x!

2,2

1,1

0,0



TRY THISTRY THIS

Find the domain, r ange and sketch the gr aph of

the following functions:

11. ( )

22. 5

1

3. 23

4.4

y f x

y g x

y h x x

x y i x

! !

! !

! !

! !

1.1 (relation,function and linear function)

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