**Determine if a relation is a function :**

Here we are going to see, how to check whether the given relation is a function or not.

Let a ∈ A and b ∈ B, then the relation "R" from A to B can be defined as follows.

R = { (a, b) }

If this relation is a function, it has to satisfy the following two conditions.

(i) Each element of A has to be involved in mapping.

(ii) Each element of A has to be mapped to at most one element in B.

In other words,

(i) Each element of A must have an image in B.

(ii) Each element of A must have an unique image (not more than one image) in B.

If either of the above conditions is not met, then the relation is not a function.

There fore,

All the functions are relations. But all the relations need not be functions.

If a relation wants to be as a function, it has to meet the above two conditions.

**Example 1 :**

In the above mapping from A to B, each element of A has an image in B. And also, each element of A unique image (not more than one image) in B.

Since, the above given relation meets both the conditions, it is a function.

**Example 2 :**

In the above mapping from P to Q, the element "c" in P does not have image in Q.

Since, the above given relation does not meet the first condition, it is not a function.

**Example 3 :**

In the above mapping from C to D, each element of C has an image in D.

But, the element 2 in C has two images (20, 40) in D, it does not meet the second condition. So, it is not a function.

Here, we are going to check, whether the given relation is a function using vertical line test.

**Key concept :**

A graph represents a function only if every vertical line intersects the graph in at most one point.

**Example 1 : **

Use the vertical line test to determine whether the following graph represents a function.

**Solution :**

Since the vertical line intersects the graph in at most one point, the given graph represents a function.

**Example 2 : **

Use the vertical line test to determine whether the following graph represents a function.

**Solution :**

Since the vertical line intersects the graph in more than one point (three points), the given graph does not represent a function.

After having gone through the stuff given above, we hope that the students would have understood , how to check whether a relation is a function.

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