Question

The function $f:N\to N$ 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

• A. one-one but not onto
• B. neither one-one nor onto
• C. onto but not one-one
• D. one-one and onto

Answer 1

The correct option is B neither one-one nor onto

$f\left(x\right)=x-5\left[\frac{x}{5}\right]$
Take $x$ in intervals of $5$ natural numbers
$f\left(x\right)=\{\begin{array}{cc}x-5\left(0\right)& 0\le x<5\\ x-5\left(1\right)& 5\le x<10\\ x-5\left(2\right)& 10\le x<15\\ x-5\left(3\right)& 15\le x<20\end{array}$
$x\in N$.
From graph, we can see that $f\left(x\right)$ is neither one-one nor onto (as it does not take all N).