Question:
Write the domain and range of function $f(x)$ given by $f(x)=\frac{1}{\sqrt{x-|x|}}$.
Solution:
Given:
f(x)=\frac{1}{\sqrt{x-|x|}}
We know that
$|x|= \begin{cases}x, & \text { if } x \geq 0 \\ -x, & \text { if } x<0\end{cases}$
$\Rightarrow x-|x|= \begin{cases}x-x=0, & \text { if } \mathrm{x} \geq 0 \\ x+x=2 x, & \text { if } \mathrm{x}<0\end{cases}$
$\Rightarrow x-|x| \leq 0$ for all $x$.
$\Rightarrow \frac{1}{\sqrt{x-|x|}}$ does not take any real values for any $x \in \mathrm{R}$.
⇒ f (x) is not defined for any x ∈ R.
Hence,
domain ( f ) = Φ and range ( f ) = Φ .