Question

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

• A. $\left(-\infty ,\infty \right)$
• B. $\left(0,\infty \right)$
• C. $\left(-\infty ,0\right)$
• D. $\left(-\infty ,\infty \right)-\left\{0\right\}$

Answer 1

The correct option is C

$\left(-\infty ,0\right)$

Explanation for the correct option(s):

Finding the domain of the function $f\left(x\right)$:

Given function $f\left(x\right)=\frac{1}{\sqrt{\left|x\right|-x}}$has to meet two conditions to be true.

1. A rational function can be true only if the denominator is $\ne 0$.
2. A square root function is always positive.

Combining the above 2 conditions $\sqrt{\left|x\right|-x}$must be greater than $0$.

$$\begin{gathered} \Rightarrow \left|x\right|-x>0 \\ \Rightarrow \left|x\right|>x \\ \Rightarrow \frac{x}{\left|x\right|}<1 \end{gathered}$$

This is possible only when $x\in \left(-\infty ,0\right)$.

Hence, the correct answer is Option (C).