f: X → Y

Function f is onto if every element of set Y has a pre-image in set X

i.e.

For every y ∈ Y,

there is x ∈ X

such that f(x) = y

 

How to check if function is onto - Method 1

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

 

Check whether the following are onto?

Example 1 - Checking one one and onto.jpg

Since all elements of set B has a pre-image in set A,

it is onto


Example 2 - Checking one one and onto.jpg

Since all elements of set B has a pre-image in set A,

it is onto


Example 3 - Checking one one and onto.jpg

Since element b has no pre-image,

it is not onto


Example 4 - Checking one one and onto.jpg

Since element e has no pre-image,

it is not onto


How to check if function is onto -  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 onto?

  1. Put y =  f(x)
  2. Find x in terms of y.

If x ∈ X, then f is onto

 

Let’s take some examples

f: R R

f(x) = x

Is f onto?

View Answer

 

f: R → R

f(x) = 1

Is f onto?

View Answer
  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 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.
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