Show that the modulus function $f:R\to R$, given by f(x) =|x|, is neither one-one nor onto, where |x| is x, if x is non -negative and |x| is -x if x, is negative.

$Here,f:R\to Risgivenbyf\left(x\right)=|x|=\{\begin{array}{c}x,ifx\ge 0\\ -x,ifx<0\end{array}$
It is seen that f(-1) =|-1|=1, f(1) =|1|=1
Therefore, f(-1) =f(1) but $-1\ne 1$
Therefore, f is not one -one.
Now, consider $-1\in R$.
It is known that f(x) =|x| is always non-negative.
Thus, there does not exist any element x in domain R such that f(x) =|x|=-1.
Therefore, f is not onto.
Hence, the modulus functions is neither one-one nor onto.