Write the domain of the real function f(x)=1 |x|-x - Vidyadip

Question

Write the domain of the real function $f(x)=\frac{1}{\sqrt{|x|-x}}$

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Answer

Case $I :$ When $x > 0.$
Then,we have, $|x|= x$
$\Rightarrow \frac{1}{\sqrt{|x|-x}}=\frac{1}{\sqrt{x-x}}=\frac{1}{0}=\infty$
Case $II :$ When $x < 0$
$| x |=- x$
$\Rightarrow \frac{1}{\sqrt{|x|-x}}=\frac{1}{\sqrt{-x-x}}=\frac{1}{\sqrt{-2 x}}(\text { exists because when } x <0,-2 x >0)$
$\Rightarrow f ( x )$ is defined when $x <0$
Therefore, domain $=(-\infty, 0)$

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