Find the range of the function,

|x| is defined as

$|\mathrm{x}|=\mathrm{x} ; \mathrm{x}>=0$

$-\mathrm{x} ; \mathrm{x}<0$

$\frac{1}{|x|}=\frac{1}{x ;} \mathrm{x}>0$

$=\frac{-1}{x} ; \mathrm{x}<0$

$\frac{x}{|x|}=1 ; \mathrm{x}>0$

$=-1 ; \mathrm{x}<0$

Hence $f(x)$ gives output values 1 and $-1$ only.

Range is $\{1,-1\}$.