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

Concept:

Domain: Domain of function f(x) is define as values of x for which function f(x) exist.

\({\rm{f}}\left( {\rm{x}} \right) = \left| {\rm{x}} \right| = {\rm{\;}}\left\{ {\begin{array}{*{20}{c}} { - x,\;\;x < 0}\\ {x,\;\;x \ge 0} \end{array}} \right.\)

Calculation:

We have to find the domain of the function \({\rm{f}}\left( {\rm{x}} \right) = \frac{1}{{\sqrt {\left| {\rm{x}} \right| - {\rm{x}}} }}\)

We know that square root is always positive.

Therefore, |x| - x > 0         (|x| - x ≠ 0)

⇒ |x| > x 

As we can see |x| is greater than x in (-∞, 0)