# HOW TO DETERMINE WHETHER THE RELATION IS A FUNCTION

## About "How to determine whether the relation is a function"

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 .

## How to check if the arrow diagram represent a function ?

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.

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