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 }.


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 = 1

y = 2(1)

y = 2

x = 2

y = 2(2)

y = 4

x = 3

y = 2(3)

y = 6

x = 4

y = 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 = 3

f(3) = 32+1

f(3) = 10

if x = 4

f(4) = 42+1

f(4) = 17

if x = 6

f(6) = 62+1

f(6) = 37

if x = 8

f(8) = 82+1

f(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

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