Question

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

Answer 1

$$\begin{gathered} \text{Case-1: When}x>0 \\ \left|x\right|=x \\ \Rightarrow \frac{1}{\sqrt{\left|x\right|-x}}=\frac{1}{\sqrt{x-x}}=\frac{1}{0}=\infty \\ \text{Case-2: When}x<0 \\ \left|x\right|=-x \\ \Rightarrow \frac{1}{\sqrt{\left|x\right|-x}}=\frac{1}{\sqrt{-x-x}}=\frac{1}{\sqrt{-2x}}\left(\text{exists because when x<0, -2x>0}\right) \\ \text{⇒}f\left(x\right)\text{is defined when}x<0 \\ \text{So, domain =}\left(-\infty ,0\right) \end{gathered}$$