A function f from the set of natural numbers to integers defined by $f\left(n\right)=\left\{\begin{array}{l}\frac{n-1}{2},\text{ when }\mathrm{n}\text{ is odd }\\ -\frac{n}{2},\text{ when n is even }\end{array}\right.$is

We have f : N$\to$$\mathrm{\mathbb{Z}}$ 

If x and y are two even natural numbers, then $f\left(x\right)=f\left(y\right)\Rightarrow \frac{-x}{2}=\frac{-y}{2}\Rightarrow x=y$

Again if x and y are two odd natural numbers then $f\left(x\right)=f\left(y\right)\Rightarrow \frac{x-1}{2}=\frac{y-1}{2}\Rightarrow x=y$

$\therefore$ f is one-one .

 Also each negative integer is an image of even natural number and each positive integer is an image of odd natural number. 

$\therefore$ f is onto.

Hence f is one one and onto both.