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 \[\left| x \right| = \begin{cases}x, & if x \geq 0 \\ - x, & if x < 0\end{cases}\] \[\Rightarrow x - \left| x \right| = \begin{cases}x - x = 0, & if x \geq 0 \\ x + x = 2x, & if x < 0\end{cases}\]

⇒ 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