The function \small f : N \rightarrow N defined by \small f(x) = x - 5\left [ \frac{x}{5} \right ]     , 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.

     

 

Answers (2)

As we learnt in

Fractional part function -

\left \{ x \right \}=x-\left [ x \right ] \right

When  \left [ x \right ]  Greatest Integer Function

 

- wherein

Range \in \left [\left 0,1 \right ) \right

 

 

 

Onto function -

If  f:A\rightarrowB 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.

-

 

 x-5\left[\frac{x}{5} \right ]=x-\frac{5}{5}[x]=x-[x]=[x]

=0\leq x<1

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}     y\epsilon N  but 0\leq x<1 so R \subset 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.

N neha

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