Question

Write the domain and range of function $f\left(x\right)=\frac{1}{\sqrt{x-|x|}}$.

Answer 1

We have,
$f\left(x\right)=\frac{1}{\sqrt{x-|x|}}$
We know that,
$|x|=\{\begin{array}{cc}x,& ifx\ge 0\\ -x& ifx<0\end{array}\Rightarrow x-|x|=\{\begin{array}{c}x-x=0,ifx\ge 0\\ x+x=2x,ifx<0\end{array}$
$\Rightarrow x-|x|\le 0$ for all x
$\Rightarrow \frac{1}{\sqrt{x-|x|}}$ does not take real values for any $xϵR$
$\Rightarrow$ f(x) is not defined for any $xϵR$
Hence, domain (f) $=\varphi$ = Range (f).