Write the domain of the real function

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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