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Q.
The domain of the function $f$ defined by $f\left(x\right)=\frac{1}{\sqrt{x-\left|x\right|}}$ is
Relations and Functions
Solution:
Given that $f\left(x\right)=\frac{1}{\sqrt{x-\left|x\right|}}$
where $x-|x| =
\begin{cases}
x-x=0, & \text{if $x\ge0$} \\[2ex]
x-(-x)=2x, & \text{if $x<0$}
\end{cases}$
Thus $\frac{1}{\sqrt{x-\left|x\right|}}$ is not defined for any $x \in R$.
Hence $f$ is not defined for any $x \in R$
i.e., domain of $f=\left\{\phi\right\}$.