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

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

it is onto Since element b has no pre-image,

it is not onto 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?

f: R → R

f(x) = 1

Is f onto?

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1. Chapter 1 Class 12 Relation and Functions
2. Concept wise
3. To prove one-one & onto (injective, surjective, bijective)

To prove one-one & onto (injective, surjective, bijective) 