Relation,Function and Linear Function

7/28/2019 Relation,Function and Linear Function

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CHAPTER 1


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Relations andFunctions


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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)xA and y B}


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Examples:

1.

2.


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Relations

A relation from A to B is any

nonempty subset of A x B .


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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)}


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Functions

Definition:

Afunction ffrom A to B is a relation from A to B

where to each a A, there corresponds exactly

one b B.

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

ordered pairs have the same first component.


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2.


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Vertical Line Test

Given the graph of a relation, ifany vertical line constructedintersects (or passes) the graphin at most one point, then the

relation described by the graphis that of a function.


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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


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The domain is also defined as the set ofall permissible values of x and

range as the set of all correspondingvalues of y.

Domain and Range


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Examples:

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

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

2.


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Functions

Two main types of functions:

Algebraic functions 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 functionsthose that are not

algebraic.


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AlgebraicFunctions


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POLYNOMIAL FUNCTIONS

General Form:

where n is a non-negative integer and

012

23

32

21

1 axaxaxaxaxaxaxPn

nn

nn

n

a0,a

1,a

2,..., a

n 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.

0na


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Standard Form:

, wherebis a real number.

Constant Functions

y f x b

Domain:Range:

fD fR b

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


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Constant Functions

Examples

Find the domain, range and sketch the

Graph of the following: 1. 2y f x

2. 3y g x


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Examples of Linear Function

4

523

x)x(h.

2. ( ) 3g x x

421 x)x(f.

Non-examples

42 . ( )

2 5i x

x

11. ( )n x

x


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How do we sketch the graph of alinear function?

Its enough to sketch a line using any 2points.

We can use

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


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Note:

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

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

Thus , if , theny-intercept: -4

x-intercept: 2

( ) 2 4y g x x

y-intercept: y=-4

x-intercept: x=2


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Relation,Function and Linear Function