**How to determine whether the relation is a function :**

If a relation has to be a function, it has to satisfy the following conditions.

(i) Domain of f is A.

(ii) For each x ∈ A, there is only one y ∈ B such that

(x, y) ∈ f .

Let us look into the following examples to understand the above concept.

**Example 1 :**

Does the following arrow diagrams represent a function? Explain.

**Solution :**

Every element in A has a unique image. Hence it is a function.

**Example 2 :**

Does the following arrow diagrams represent a function? Explain.

**Solution :**

C has two images namely 20 and 40. Hence, it is not a function.

**Example 3 :**

Let X = { 1, 2, 3, 4 }. Examine whether each of the relations given below is a function from X to X or not. Explain

f = { (2, 3), (1, 4), (2, 1), (3, 2), (4, 4) }

**Solution :**

To check whether the relation forms a function, it has to satisfy the above two conditions.

(i) Domain of f = {1, 2, 3, 4} = Set X

(ii) Every element in X must be associated with different elements. Here 2 is associated with two different elements 3 and 1.

Hence it is not a function.

**Example 4 :**

Which of the following relations are functions from A = { 1, 4, 9, 16 } to B = { –1, 2, –3, –4, 5, 6 }? In case of a function, write down its range.

f = { (1, –1), (4, 2), (9, –3), (16, –4) }

**Solution :**

To check whether the relation forms a function, it has to satisfy the above two conditions.

(i) Domain of f = {1, 4, 9, 16} = Set A

(ii) Each element in A is associated with elements in B.

Hence it is not a function.

Range = {-1, 2, -3, -4}

**Example 5 :**

If X = { 1, 2, 3, 4, 5 }, Y = { 1, 3, 5, 7, 9 } determine which of the following relations from X to Y are functions? Give reason for your answer. In case of a function, write down its domain, range and codomain.

R = { (1, 1), (2, 1), (3, 3), (4, 3), (5, 5) }

**Solution :**

To check whether the relation is a function, it has to satisfy the above two conditions.

(i) Domain of R = {1, 2, 3, 4, 5} = Set X

(ii) Every element in X has image in Y.

Hence it is not a function.

Domain = {1, 2, 3, 4, 5}

Range = {1, 3, 5}

Co domain = { 1, 3, 5, 7, 9 }

**Example 6 :**

Let A = { 10, 11, 12, 13, 14 }; B = { 0, 1, 2, 3, 5 } determine the following relations from X to Y are functions? Give reason for your answer.

f = { (10, 1), (11, 2), (12, 3), (13, 5), (14, 3) }

**Solution :**

To check whether the relation forms a function, it has to satisfy the above two conditions.

(i) Domain of f = {10, 11, 12, 13, 14} = Set A

(ii) Every element in A has image in B.

Hence it is not a function.

Let us see the next example on "How to determine whether the relation is a function".

**Example 7 :**

Does the following arrow diagrams represent a function? Explain.

**Solution :**

Every element in P has a unique image. Hence it is a function.

**Example 8 :**

Does the following arrow diagrams represent a function? Explain.

**Solution :**

Every element in P has a unique image. Hence it is a function.

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

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