Question:
Write the domain of the real function $f(x)=\frac{1}{\sqrt{|x|-x}}$.
Solution:
Case-1: When $x>0$
$|x|=x$
$\Rightarrow \frac{1}{\sqrt{|x|-x}}=\frac{1}{\sqrt{x-x}}=\frac{1}{0}=\infty$
Case-2: When $x<0$
$|x|=-x$
$\Rightarrow \frac{1}{\sqrt{|x|-x}}=\frac{1}{\sqrt{-x-x}}=\frac{1}{\sqrt{-2 x}}$ (exists because when $x<0,-2 x>0$ )
$\Rightarrow f(x)$ is defined when $x<0$
So, domain $=(-\infty, 0)$
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