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
__

**How to check if function is**

**onto -**

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?

- Put y = f(x)
- 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?

-a-

We follow the steps

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

If x ∈ X, then f is onto

y = f(x)

y = x

∴ x = y

Since y ∈
**
R
**

x = y also belongs to R

i.e. x ∈
**
R
**

**
∴ f is onto
**

-ea-

f:
**
R → R
**

f(x) = 1

Is f onto?

-a-

f(x) = 1

∴ y = 1

So, value of y will always be 1

∴ There is no value x where y = 2

⇒ 2 does not have a pre-image in X

∴ f is
**
not onto
**

-ea-