Write the domain and range of function f(x) given by f(x)

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 ( ) = Φ .

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