Find the domain of the following functions:\(f(x)=\frac{1}{\sqrt{x-|x|

1.	Find the domain of the following functions:\(f(x)=\frac{1}{\sqrt{x-\|x\|}}\)
Answer» Given,\(f(x)\frac{1}{\sqrt{x-\|x\|}}\) We know that, \|x\| = \(\begin{cases}x, & \quad \text{when } x \geq0\\-x, & \quad \text{when } x<0\end{cases}\) x - \|x\| = \( \begin{cases}x-x & \quad \text{when } x\geq0\\x+x, & \quad \text{when } x<0\end{cases}\) x - \|x\| = \(\begin{cases}0, & \quad \text{when } x\geq0\\2x, & \quad \text{when } x <0\end{cases}\) = x − \|x\| ≤ 0 For all x \(\frac{1}{\sqrt{x-\|x\|}}\) does not take real values for any x ∈ R 1 ⇒ f(x) is not defined for any x ∈ R Hence, Domain \(f(x)=\phi\)

Find the domain of the following functions:\(f(x)=\frac{1}{\sqrt{x-|x|}}\)

Answer»

Given,\(f(x)\frac{1}{\sqrt{x-|x|}}\)

We know that,

|x| = \(\begin{cases}x, & \quad \text{when } x \geq0\\-x, & \quad \text{when } x<0\end{cases}\)

x - |x| = \( \begin{cases}x-x & \quad \text{when } x\geq0\\x+x, & \quad \text{when } x<0\end{cases}\)

x - |x| = \(\begin{cases}0, & \quad \text{when } x\geq0\\2x, & \quad \text{when } x <0\end{cases}\)

= x − |x| ≤ 0 For all x

\(\frac{1}{\sqrt{x-|x|}}\) does not take real values for any x ∈ R 1

⇒ f(x) is not defined for any x ∈ R Hence, Domain \(f(x)=\phi\)