# 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.

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

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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