f: X → Y

Function f is one-one if every element has a unique image,

i.e.

when f(x 1 ) = f(x 2 )

⇒ x 1 = x 2

Otherwise the function is many-one.

 

How to check if function is one-one - Method 1

In this method, we check for each and every element manually if it has unique image

 

Check whether the following are one-one ?

1.jpg

Element

Image

1

a

2

b

3

c

4

d

Since every element has a unique image,

it is one-one     

 


One One function - Part 2


Element

Image

1

b

2

c

3

d

4

a

Since every element has a unique image,

it is one-one    


One One function - Part 3

Element

Image

1

a

2

a

3

c

4

d

Since 1 and 2 has same image,

it is not one-one


One One function - Part 4

Element

Image

1

a

2

b

3

c

4

d

Since every element has a unique image,

it is one-one     

 


How to check if function is one-one -  Method 2

 

This method is used if there are large numbers

 

Example:

f : N N    (There are infinite number of natural numbers)

f : R R    (There are infinite number of real numbers)

f : Z Z     (There are infinite number of integers)

 

Steps :

How to check one-one?

  1. Calculate f(x 1 )   
  2. Calculate f(x 2 )   
  3. Put f(x 1 ) = f(x 2 )  

If x 1 = x 2 , then it is one-one.

Otherwise, many-one

 

Let’s take some examples

f: R R

f(x) = x

Is f one-one?

-a-

We follow the steps

  1. Calculate f(x 1 )
  2. Calculate f(x 2 )
  3. Put f(x 1 ) = f(x 2 ),
  4. If x 1 = x 2 , then it is one-one.
    Otherwise, many-one

 

f(x 1 ) = x 1

f(x 2 ) = x 2

Putting f(x 1 ) = f(x 2 )

x 1 = x 2

 

Since x 1 = x 2 ,

f is one-one.

-ea-

 

f: R R

f(x) = 1

Is f one-one?

-a-

Since

One One function - Part 5

f is not one-one.

-ea-

  1. Chapter 1 Class 12 Relation and Functions
  2. Concept wise

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.