Show that the Modulus Function
Question:

Show that the Modulus Function $f: \mathbf{R} \rightarrow \mathbf{R}$ given by $f(x)=|x|$, is neither one-one nor onto, where $|x|$ is $x$, if $x$ is positive or 0 and $|x|$ is $-x$, if $x$ is negative.

Solution:

$f: \mathbf{R} \rightarrow \mathbf{R}$ is given by,

$f(x)=|x|=\left\{\begin{array}{l}x, \text { if } x \geq 0 \\ -x, \text { if } x<0\end{array}\right.$

It is seen that $f(-1)=|-1|=1, f(1)=|1|=1$.

$\therefore f(-1)=f(1)$, but $-1 \neq 1$.

$\therefore f$ is not one-one.

Now, consider $-1 \in \mathbf{R}$.

It is known that $f(x)=|x|$ is always non-negative. Thus, there does not exist any element $x$ in domain $\mathbf{R}$ such that $f(x)=|x|=-1$.

∴ f is not onto.

Hence, the modulus function is neither one-one nor onto.