# HOW TO CHECK IF EACH RELATION IS A FUNCTION

How to Check If Each Relation is a Function :

Here we are going to see "How to Check If Each Relation is a Function".

## How to Check If Each Relation is a Function

A relation f between two non-empty sets X and Y is called a function from X to Y if, for each x  X there exists only one y Y such that (x, y)  f .

That is, f = {(x,y)| for all x  X, y ∈ Y }.

## How to check if the given relation is a function using arrow diagram This represents a function. Each input corresponds to a single output. This represents a function. Each input corresponds to a single output.

Question 1 :

Let f = {(x, y) | x, y  N and y = 2x} be a relation on ℕ. Find the domain, co-domain and range. Is this relation a function?

Solution :

Since x and y  N,

 x = 1y = 2(1)y = 2 x = 2y = 2(2)y = 4 x = 3y = 2(3)y = 6 x = 4y = 2(4)y = 8

f = {(1, 2) (2, 4) (3, 6) (4, 8)................}

For each values of x, we get different values of y. So the given relation is a function.

Domain is the set of values of x

Domain  =  {1, 2, 3, 4, ............}

Co domain is the set of value of y. Since  N

Co domain  =  {1, 2, 3, 4, .............}

Range means the set of values of y, which are associated with x.

Range =  {2, 4, 6, 8, .....}

Question 2 :

Let X = {3, 4, 6, 8}. Determine whether the relation ℝ  = {(x, f (x)) | x  X, f (x) = x2 + 1} is a function from X to  ?

Solution :

Given that :

f (x) = x2 + 1

x  X

 if x = 3f(3) = 32+1f(3) = 10 if x = 4f(4) = 42+1f(4) = 17 if x = 6f(6) = 62+1f(6) = 37 if x = 8f(8) = 82+1f(8) = 65

R  =  { (3, 10) (4, 17) (6, 37) (8, 65) }

For each values of x, we get different values of f(x).

Hence it is a function After having gone through the stuff given above, we hope that the students would have understood, "How to Check If Each Relation is a Function".

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