Question

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

Answer 1

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

We know that
$\left|x\right|=\left\{\begin{array}{ll}x,& \mathrm{if}x\ge 0\\ -x,& \mathrm{if}x<0\end{array}\right.$
$\Rightarrow x-\left|x\right|=\left\{\begin{array}{ll}x-x=0,& \mathrm{if}\mathrm{x}\ge 0\\ x+x=2x,& \mathrm{if}\mathrm{x}<0\end{array}\right.$
⇒ x $-$ | x| ≤ 0 for all x.
$\Rightarrow \frac{1}{\sqrt{x-\left|x\right|}}$ does not take any real values for any x ∈ R.
⇒ f (x) is not defined for any x ∈ R.

Hence,
domain ( f ) = Φ and range ( f ) = Φ .