The function f: N → I defined by f(x) = x - 5 [(x)/(5)] , where N...

Given, $f( x ) = x - 5[ \dfrac{x}{5} ]$

Hence, $f( x ) = 5( \dfrac{x}{5} - [ \dfrac{x}{5} ] )$

$= 5{ \dfrac{x}{5} }$

Since, $x$ is a natural number.

Hence, \left\{\dfrac{x}{5}\right\} = 0, \dfrac{1}{5}, \dfrac{2}{5}, \dfrac{3}{5}, \dfrac{4}{5}$<br /><br />Hence,$f( x ) = 0$, when$x = 5n, n \in N$<br /><br />$= 1$when$x = 5n + 1, n \in N$<br /><br />$= 2$when$x = 5n + 2, n \in N$<br /><br />$= 3$when$x = 5n + 3, n \in N$<br /><br />$= 4$when$x = 5n + 4, n \in N$<br /><br />Hence,$f\left(x\right) is neither one-one nor onto.