Question

For $f$ to be a function, the domain of $f$, if $f\left(x\right)=\frac{1}{\sqrt{\left|x\right|-x}}$ is

• A. $(-\infty ,0)$
• B. $(0,\infty )$
• C. $(-\infty ,\infty )$
• D. $(-\infty ,0]$

Answer 1

The correct option is A $(-\infty ,0)$

Given $f\left(x\right)=\frac{1}{\sqrt{|x|-x}}$

We know $f\left(x\right)$ is defined if $|x|-x>0$

$\Rightarrow |x|>x$

For $x>0,x>x$ (not possible) and

For $x<0,-x>x\Rightarrow 2x<0\Rightarrow x<0$ (possible)

Hence domain of $f$ is $(-\infty ,0)$