lecture 1.1 real no , relation & functions

8/7/2019 Lecture 1.1 Real No , Relation & Functions

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1-1: Real Numbers,

Relations & Functions

Learning Goals:Rational & irrationalnumbers

coordinate system in aplane.concept of relations &

functions


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Definition

The Real Number System:

Real Numbers

Rational

Numbers

Irrational

Numbers


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Definition

Rational Numbers can beexpressed in the form

r

s

where r and s are integersand 0s { . Rational numbers

can be written as a decimalthat either terminates or

repeats in a pattern.


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Example

Rational Numbers:

1

4

r

0s {

5

-3

2

3

7

3

5560


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Important Idea

0 1-4 57

2

A rational number is anumber that can befound on a number line.


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Definition

Irrational Numbers are

numbers that cannot be

expressed in the form

and cannot be found on anumber line.

r

s


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Example

Irrational Numbers:

2

5

95

T

e


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Important Idea

Any number that cannot bewritten as a decimal thateither terminates or repeatsin a pattern is an irrational

number. Square roots ofnumbers that are not perfect

squares are always irrational.


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Important Idea

Numbers such as arecalled irrational numbers

and are always betweentwo other numbers.

2

1 2

1.51.4

2


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...If is always between

two numbers, whereexactly is it?

2

Could it be that it doesntexist ???

Important Idea


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Try This

True or False:

9 is an irrational number.

False


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Definition

CartesianCoordinate

System:

III

III IV

quadrants

axes

origin


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ExamplePlot thefollowingordered pairs:

(0,5)

(2,4)

(3,-5)

(-4,-4)


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Definition

The Domain is all thepossible values thatx canhave. The Range is all thepossible values thaty can

have.f(x) is another name for y.


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Example

The table shows the height andarm span in cm. of six membersof the cheerleading squad:

Height 178 189 158 169 195 188

Arm

Span

180 180 163 169 189 182

Write the relations domain andrange using set notation.


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Example

Given the relation{(0,1),(1,2), (-1,2),(2,5),(-

2,5)(3,10),(-3,10)}, state itsdomain & range, create ascatter plot and find a rulethat relates the value of thefirst coordinate to the value

of the second coordinate.


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Try This

Given the relation{(0,0),(1,1),(1,-1),(4,2),(4,-

2)(9,3),(9,-3)}, state itsdomain & range, create ascatter plot and find a rulethat relates the value of thefirst coordinate to the value

of the second coordinate.


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Solution

Domain={0,1,4,9}Range={-3,-2,-1,0,1,2,3}

Rule: 2x y! or y x! s

Scatter Plot:Lists:


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Definition

Afunction is a set of orderedpairs such that the firstcoordinate denotes the input,the second coordinatedenotes the output that is

obtained from th

e rule of th

efunction, and each inputcorresponds to one and only

one output.


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Important IdeaA function is likea meat grinder.

You putsomething in(Domain), turnthe crank(implement the

Rule), andcollect

the output (Range).


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Example

In the following sets of orderedpairs, the first coordinate

represents an input and the

second coordinate representsthe corresponding output.Explain why each set is, or is

not, a function.a) {(0,1),(-1,3),(2,4),(2,-1)}

b) {(0,1),(-1,3),(2,4),(3,-1)}


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Try This

In the following set of orderedpairs, the first coordinate

represents an input and the

second coordinate representsthe corresponding output.Explain why each set is, or is

not, a function.a) {(0,1),(1,3),(2,4),(3,5)}

b) {(0,1),(2,3),(2,4),(3,5)}


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Example

These areexamples of

functions. Eachinput musthave

one and only

one output.{(0,1),(1,3),(2,4),(3,5)}


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These areexamples ofrelations which

are notfunctions. Eachinputhas morethan oneoutput.

{(0,1),(1,3),(1,4),(3,5)}

Example


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Definition

Function Notation: Instead of2 3y x! , we write

2( ) 3. f x x! If the input is 2,

then t

he output is 7, t

hat is :( 2 ) 7 .f ! The rule is: 2 3x


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Try ThisFor the function: ( ) f x x!

What is ,(4)f (9)f ( 4)?f &

What is the domain (inputs) andrange (outputs)?

(4) 2f !(9) 3f !

( 4)f is undefined

D= { }x ouR={ 0}y u

lecture 1.1 real no , relation & functions

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