#### The function defined by     , where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is : Option 1) one-one and onto. Option 2)  one-one but not onto. Option 3)  onto but not one-one. Option 4)  neither one-one nor onto.

As we learnt in

Fractional part function -

When    Greatest Integer Function

- wherein

Onto function -

If  f:AB is such that each & every element in B is the f image of atleast one element in A.Then it is Onto function.

- wherein

The range of f is equal to Co - domain of f.

One - One or Injective function -

A line parallel to x - axis cut the curve at most one point.

-

So that for difference value of  x - [x]

lies between 0 and 1 . [0, 6) so that this is not one - one.

Now, y = {x}       but  so  co-domain. So that this is not onto.

Correct option is 4.

Option 1)

one-one and onto.

This is an incorrect option.

Option 2)

one-one but not onto.

This is an incorrect option.

Option 3)

onto but not one-one.

This is an incorrect option.

Option 4)

neither one-one nor onto.

This is the correct option.