ma thematic presentation chapter relation and function

8/8/2019 Ma Thematic Presentation Chapter Relation and Function

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Relation And Function

Annisa Fitriana (8 SBI B/2)

Dzaky El Fikri (8 SBI B/12)

Askari Ishmat (8 SBI B/17)

Wahyu Nugroho (8 SBI B/18)


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Domain and range, relations andfunctions

Relation:A rule for pairing the elements of set a withthe element of set b.

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

Domain: all of the first coordinates of a relation (thinkof the domain as the inputs).

3,1,0

Range: all of the 2nd coordinates of a relation (thinkof the range as the outputs).

2, 5, 2

Function: a relation in which each element of the


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{(2,3),(-1,5),(4,-2),(9,9),(0,6)} This is arelation

The domain is the set of all x values in the relation

{(2,3),(-1,5),(4,-2),(9,9),(0,6)}

The range is the set of all y values in the relation

{(2,3),(-1,5),(4,-2),(9,9),(0,6)}

domain = {-1,0,2,4,9}

These are example of the x values written in a set from

smallest to largest

range = {-6,-2,3,5,9}

These are the y values written in a set from smallest to

largest

DOMAIN AND RANGE


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Domain (set of all xs) Range (set of all ys)

1

2

3

4

5

2

10

8

6

4

A relation assigns the xs with ys

This relation can be written {(1,6),(2,2),(3,4),(4,8),(5,10)}


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Ow now I know

about Function

Set A is the domain

1

2

3

4

5

Set B is the range

2

10

86

4

Must use all the xs

A function f from set A to set B is a rule of correspondence that assigns to each element

x in the set A exactly one element y in the set B.

The x value can only be assigned to one y


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Menu

Understanding relation. Arrow diagram.

Cartesian diagram.

Set of ordered pairs.

Function.

Notation.


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Understanding Relation

Relation from set A to set B is defined as a connector to pair themembers of set A to the members of set B.

For example, set A {Konci, Reza, Nosa} can be connected to setB {Ahmad, Aji, Heri} by relation son of.

Relations can be expressed by 3 ways, arrow diagram,Cartesian diagram, and set of ordered pairs.


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ArrowDiagram

Arrow diagram uses 2 sides to displays the members of set Aand set B.

These members of 2 sets are connected each other by arrows,

and these arrows represents the relation of both sets.

The relation is written between the set names.

Format

Example


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ArrowDiagram Format

AA BBRelationRelation


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Konci

Reza

Nosa

Example of ArrowDiagram

Aji

Ahmad

Heri

AA BBis the son ofis the son of


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CartesianDiagram

This diagram is usually used in determiningcoordinates, but also useful for expressing relation.

This diagram uses 2 axis's, x axis and y axis. X axis

usually used for domain, and y axis for codomain.

Example


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CartesianDiagram Example


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Set ofOrdered Pairs

It is used by listing a member and its pair in a bracket. For example: {(Konci, Ahmad), (Nosa, Heri), (Reza,

Aji)}.


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Function

For example, set A contains name of boys (Nosa, Suluh, Konci,Reza), and set B contains name of biologist fathers (Heri, Budi,

Aji, and Akhmad).

Because that, we can build a relation between two sets whichevery element of set A always have exactly one pair in set B,

because a child can only able to have one biologist father.T

hisspecial kind of relation is called Function.

Function is special relation while the members in domain hasexactly one pair in codomain.


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Terms in Function

Domain:The first region in function. All of the memberof set in this region must have exactly one pair in

codomain to be a function.

Codomain:The region of pair set of domain.

Range:The region of members of pair set of domain.

Range is always become region of codomain, but

codomain isnt always become region ofRange.

To count possible ways of function of both sets,

n(codomain)n(domain)


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Notation of Function

Notation of function is a notation that represents therelation of that both sets including the characteristicof that set.

If the members of domain set is called x, the notationof members of domain set is called f(x).

If the function from domain set to codomain set isless 1 than, we can write the notation of functionf(x)=x+1


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One toOneCorrespondence

A set is said to have A one to one correspondence with set B

if every element of A is paired with exactly one element of B

and every element of set B is paired with exactly one

element of A.T

herefore the number of elements of the set Aand B must be equal


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Function Formula

Formulating function

It is common to introduce a function which follow a certain

rules with a name such as f , g , h or any other lower case

Latin letters

A mapping by a function f hat map ever element x of set A to

element y of set B can be denoted

the notation f : x y is read function f maps x to y there y

is called the image (map) of x under y

F : x y


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Figure below shows a function f mappng A to B if x is an

elements of the doain of A then the image of x under f is

denoted by f(x) and is read a function of x

Example given the mapping f : x x + 2

Since the image of x under f can be denoted by f(x) we can

express the mapping as f(x) = x + 2

The form f(x) = x + 2 is called a function formula

x F(x)


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Number Possible Ways

ofOne toOne

Correspondence

A = { a , b , c } B = { 1 , 2 , 3 }

To find how many possible ways one to one correspondence

set A to set B

N(A) = N(B) = 3!(three factorial)

= 3.2.1 = 6 possible ways


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We commonly call functions by letters. Because function starts with f, it is a commonlyused letter to refer to functions.

632 2 ! xxxf

The left hand side of this equation is the function notation. It tells us two things. We

called the function f and the variable in the function is x.

This means the

right hand side

is a function

called f This means the

right hand side has

the variable x in it

The left side DOES NOT MEAN f

times x like parenthesis usually do, it

simply tells us what is on the right

hand side.


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632 2 ! xxxf

So we have a function called f that has the variable x in it.

Using function notation we could then ask the following:

623222 2 !f

8668623422 !!!f

Dont forget order of operations---powers, then multiplication, finally

addition & subtraction

f(2). This means to find the function f and instead of having an x in it,put a 2 in it. So lets take the function above and make parenthesis

everywhere the x was and in its place, put in a 2.


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6322

!

xxxf

Example f(-2).

623222 2 !f

20668623422 !!!fExample f(3). 632 2 ! xxxf

633323 2 !f

362718693923 !!!f


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532 ! xxxf

55355 2 !f

4551525653255 !!!f

Example f(5).


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63 ! xxfThis means the

right hand side

is a function

called f This means the

right hand side has

the variable x in it,

If the variable x

change like x+3 orx+4 the right side

also change the

variable x with the

command.

If the variable change

This x has to be

changed into thevalue of x in the left

example: in the left

show x+3, we have

to changed this x

into the x+3 to.

Untuk hasil

perubahan

fungsi adalah

dengan

mengurangkan

hasil fungsi

dengan f(x)

contoh:

f(x+3)-f(x)= 15

The first step find th

result of f(x+3)

The second

subtract the result

of f(x+3) with f(x)


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53 ! xxf

5333 ! xxf

1435933 !! xxxf

Example f(x+3).


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4! xxf

444 ! xxf

8444 !! xxxf

Example f(x+4).


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4! xxf

41515 ! xxf

5541515 !! xxxf

Example f(5x+1).


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4! xxf

41515 ! xxf

5541515 !! xxxf

Example f(5x+1).


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4! xxf

5)1(21 ! xxf

5)1(21 ! xxf

Example f(x+1).

725221 !! xxxf


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f(x+1)-f(x)= 12

The first step findthe result of f(x+1)

and the f(x) is 2x+5.

The second

subtract the result

of f(x+3) with f(x)

To know the change result of he f(x)

we subtract the result of the function

with f(x) example

52 ! xxf

5)1(21 ! xxf

5)1(21 ! xxf

725221 !! xxxf

5272)()3( ! xxxfxf

125722)()3( !! xxxfxf

12)()3( ! xfxf12 is the value of the change of

function


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5! xxf

533 ! xxf

8533 !! xxxf

Example f(x+3).

13 is the value of the change of

function

58)3( ! xxxfxf

the value of the change of function

1358)3( !! xxxfxf


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7! xxf

744 ! xxf

11744 !! xxxf

Example f(x+4).


function

711)3( ! xxxfxf


18711)3( !! xxxfxf


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4! xxf

433 ! xxf

7433 !! xxxf

Example f(x+3).


function

48)3( ! xxxfxf


1257)3( !! xxxfxf


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One to one correspondence

The sum of element of set A and B must same.

A set A is said to have a one to one correspondence

with set b if every element of set A is paired with

exactly one element of A, therefore the number

element of set a and must be equal Number possible way of the one to one

correspondence:

Using factorial number(!)

Example: A={a,b,c} B={1,2,3}. Answer: 3!=3.2.1=66 is the number possible of the one to one correspondence

between set A to set B


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Function Graph

Graph of function is special relation from set A to set

B in form of graph.The form of graph is Cartesian

Diagram.


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How to make Function Graph

First we make table like this :

And Finally make the Cartesian Diagram from the table

x

f(x)

x,f(x)


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Example of Function Graph

f (x) = x + 5

x = {1,2,3,4,5)


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Example of Function Graph


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Table

x

f(x)

x,f(x)

1 2 3 4 5

109876

(1,6) (2,7) (3,8) (4,9) (5,10)


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CartesianDiagram

1 4 532

6

10

9

8

7


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ma thematic presentation chapter relation and function

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