My IntroductionName: Prakash ADHIKARI

Academic Qualification:Master’s Degree Research, Universite Lumiere-IUT Lumiere, Lyon - 2(ULL-2), France Master’s Degree in Education Majoring Mathematics, TU NepalB.ED Mathematics - TU, Nepal

Teaching Background:Mathematics Instructor, Heartland Academy,Kathmandu 2010 to 2013

Relation and Function

Relation and Function

1. Cartesian Product2. Relation3. Function

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How many ways each player of one team handshake with players of another team?

Germany (A) Argentina (B)

KevinManuelLukas

MessiRomerozabaleta

Cartesian Product

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A×B = {(kevin,Messi),(Kevin,Romero), (Kevin,zabaleta),(Manuel,Messi),(Manuel,Romero),(Manuel, Zabaleta),(Lukas,Messi),(Lukas,Romero),

(Lukas,Zabaleta)}

Kevin

Manuel

Lukas

Messi

Romero

Zabaleta

A B

1 to 1 many to many

Relations

1 to many

Countries

UK

Nepal

India

Bangladesh

London

Dhaka

New Delhi

Paris

Kathmandu

Relation

Capital Cities

France

Relation - ‘ is the capital city of’

1

2

3

4

5

2

10

8

6

4

Relation

This relation is R= {(1,6), (2,2), (3,4), (4,8), (5,10)}

A B

This is a relation

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

Domain = {-1,0,2,4,9} All x Values

Range = {-6,-2,3,5,9} All y Values

Relations

Domain and Range in Relation

Function

Rice Peeling and Milling Machine f(x) means Function of x

Function

In a Function, One Input ALWAYS has exactly one output

2 x3 8

• 2 is the input number (or x-value on a graph).• 8 is the output number (or y-value on a graph).

• The illustrates the idea of a function.

Function

Input Output

x3

‘f ’ is said to be a FUNCTION from a set A to a set B , If Each element a єA associates with a unique element in bє B.

This means there must be at least one arrow (no more) leaving each point in the domain.

We write Where each element a єA is

uniquely associated to b B єso that, f (a) = b

Definition: Function

f

Then, we say that Set A is the Domain of f

Set B is the Co-domain of f

If f(a) = b, we say that bєB is the Image of a

aєA is the Pre-image of b.

The Range of f:AB

is the set of all images

of elements of A.

Functions: Domain, Codomain and Range

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Consider an example with Set A= { 1,2,3,4} and B={3,6,11,18} We have function f (x) = x2+2 from A to B ;

Domain of f ={ 1,2,3,4}Then Range of f = {3,6,11,18}

Functions: Domain, Codomain and Range

15: {3,6,11,18}

: {1,2,3,4}

f(x)= x2+2

Given f(x) = 4x + 8, find each:1. f(2)

2. f(a +1) = ?

3. f(4a) = ?

Evaluating Functions

= 4(2) + 8= 16

If f(x) = 3x 1, and g(x) = 5x + 3, find each:

Evaluating More Functions

= ?

= ?

1. f(2) + g(3)

2. f(4) - g(-2)

3. 3f(1) + 2g(2)

How to Know the given Relation is function or not???

Functions

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Input number Output number

Can you have one letter going to two different mail boxes???

Not a FUNCTION

To understand in Better way:

Input number

Output number

Can you have two different letters going to one mail box?

Can you have a letter going to one mail box?

More Ideas..

Voters Candidates

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1.

f is a function f is a function

2.

1234234

AA

B

1234

A B

361118234

1234234

1234

BA AA BA

f is not a function f is not a function

3. 4.

wxyz

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A B A B

1234234

361118234

1234234

36114

AA BA AA BA

• One-to-one (or injective)• Many to One• Onto ( Or Surjective)

Types of Function

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1. One to One Function•f is one-to-one (or injective) function, if and only if it does not map two distinct elements of A onto the same element of B. •In other words:A function f:AB is said to be one-to-one iff, x, yA (f(x) = f(y) x = y)

Types of Function

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f1234

1234

A B

2. Many to One Function

•Association of more than one element of Domain with single element in Range.• f(1) = 1, f(2) = 1, f(3) = 1 and f(4) = 1 So f is Many-to-one function.

Types of Functions

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f

1234

1

A B

3. Onto Function:•A function f:AB is called onto (or surjective) function if and only if for every element bB there is an element aA with f(a) = b.•In other words,

If the codomain set is equal to range set then the function is onto or Surjective.

Types of Functions

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Codomain = { 1,4,9)Range of Function = {1,4,9)

CODOMAIN = RANGE

Types of Function

4. Identity Function

• Let A be any non- empty set, • The function defined by i(a)=a for all

aєA, is called Identity Function of Set A• Example: Let A= {1,2,3,4} then is given by

i(1)=1i(2)=2i(3)=3i(4)=4 28

𝒊 : 𝑨→ 𝑨

𝒊 : 𝑨→ 𝑨

Discussing questions:• Define One-to-One Function with example.• What is Onto Function? Give an example.

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• An inverse function is a Function that "reverses" another function: if the function f applied to an input x gives a result of y under f, then applying its inverse function f-1 to y gives the result x

i.e. f(x) = y Iff f-1(y) = x

INVERSE OF FUNCTION

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y x

f

f-1

f(x)=y

f-1(y)=x

An Example of INVERSE FUNCTION

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f(a) = 3f(b) = 1f(c) = 2

f-1 (3) = af-1 (1) = bf-1 (2) = c

f-1:CP is no function, because it is not defined for all elements of C and assigns two images to the pre-image New York.

INVERSE OF FUNCTION

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Linda

Max

Kathy

Peter

Boston

New York

Hong Kong

Moscow

LübeckHelena

f

f-1

Inverse of a function MAY NOT BE A FUNCTION

Inverse Function• Example:• If f(x)= 4x+5, then find ) and Solution: f(x) can be written as y, Therefore, y= 4x+5 Changing domain as range and vice versa Ans Ans

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Composition or Composite Functions

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•The composition ( or Composite of two functions) f:AB and g:BC, denoted by g0f, is defined by

[(g0f)a] = g(f(a))

Composition

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•Example:

• f(x) = 7x – 4, g(x) = 3x,•Function is defined as f:RR, g:RR

•(gof)(x) = g(f(x)) = g(7x-4) = 3(7x-4) = 21x - 12•If x=5,•(gof)(5) = g(f(5)) = g(31) = 93

Composition

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• Composition of a function and its inverse:

(f-1f)(x) = f-1(f(x)) = x•The composition of a function and its inverse is the

identity function i(x) = x.

Composition

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• Given the function f(x)= x-6, xєR, Find the values of :

i. f-1(x) ii. f-1(12) • Given the function f(x)= 4x+9, xєR, Find the

values of :

i. ff(x) ii. fff(x) • Given Functions are f(x)= 4x+9 ; g(x)= x2+1, xєR,

Find the values of:

i. f-1(x) ii. fo(g)x and iii. gof(x)

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Work on paper, Now you try….!

Additional Questions• Functions f and g are defined by: f:x→2x+3 and

g:x→x2-6x– Express f-1(x) in terms of x.– Solve the equation if f(x)=f-1(x)– Find f-1g(x)

• The functions f and g are defined for xєR by f:x→3x+a and g:x→b-2x Where a and b are constants.

• Given that ff(2)=10 and g-1(2)=3, - Find the values of a and b.- An expression for fg(x)

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Thank You

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