Question

Show that the Signum function $f:R\to R$, given by

$f\left(x\right)=\{\begin{array}{c}1,ifx>0\\ 0,ifx=0\\ -1,ifx<0\end{array}$
is neither one-one nor onto.

Answer 1

$f\left(x\right)=\left\{\begin{array}{cc}1& \text{if}x>0\\ 0& \text{if}x=0\\ -1& \text{if}x<0\end{array}\right\}$.

It is seen that $f\left(1\right)=f\left(2\right)=1$, but $1\ne 2$.

$\therefore f$ is not one-one.
Now, as $f\left(x\right)$ takes only $3$ values $(1,0,$ or$-1).$

For the element $-2$ in co-domain $R$, there does not exist any $x$ in domain $R$ such that $f\left(x\right)=-2$
$\therefore f$ is not onto
Hence, the signum function is neither one-one nor onto.