The function f : N → N defined by f\left ( x \right ) = x -5 \left [ \frac{x}{5} \right ] , where N is the set of natural numbers and \left [ x \right ]  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 (1)

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

Taking x in an interval of five natural numbers, we have the following: 

f(x)=\left\{\begin{array}{ll} x-5(0), & 0 \leq x < 5 \\ x-5(1), & 5 \leq x < 10 \\ x-5(2), & 10 \leq x < 15 \\ x-5(3), & 15 \leq x < 20 \end{array}\right.

Therefore, here, x ∈N. hence, f(x) is neither a one-one function nor onto function.

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