**How to determine if the function is onto ?**

Here we are going to see how to determine if the function is onto.

**Definition of onto function :**

A function f : A -> B is said to be an onto function if every element in B has a pre-image in A. That is, a function f is onto if for each b ∊ B, there is atleast one element a ∊ A, such that f(a) = b.

This is same as saying that B is the range of f . An onto function is also called a surjective function. In the above figure, f is an onto function

Let us look into some example problems to understand the above concepts.

**Example 1 :**

Check whether the following function is onto

f : N → N defined by f(n) = n + 2.

**Solution :**

**Domain and co-domains are containing a set of all natural numbers. **

**If x = 1, then f(1) = 1 + 2 = 3 **

**If x = 2, then f(2) = 2 + 2 = 4**

**From this we come to know that every elements of codomain except 1 and 2 are having pre image with. **

**In order to prove the given function as onto, we must satisfy the condition **

**Co-domain of the function = range **

**Since the given question does not satisfy the above condition, it is not onto.**

**Example 2 :**

Check whether the following function is onto

f : R → R defined by f(n) = n^{2}

**Solution :**

**Domain = All real numbers **

**Co-domain = All real numbers**

**Since negative numbers and non perfect squares are not having preimage. It is not onto function.**

**Example 3 :**

Check whether the following function are one-to-one

f : R - {0} → R defined by f(x) = 1/x

**Solution :**

**Domain = all real numbers except 0**

**Co-domain = All real numbers including zero.**

**In co-domain all real numbers are having pre-image. But zero is not having preimage, it is not onto.**

After having gone through the stuff given above, we hope that the students would have understood "How to determine if the function is onto".

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