Find the domain of the following functions: f(x) = 1/ √x+|x|

Question

Find the domain of the following functions:

\(f(x)=\frac{1}{\sqrt{x+|x|}}\)

Answer 1

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

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

Now, \(f(x)=\frac{1}{\sqrt{x+|x|}}\)assume real values, if 

x + |x| > 0 

⇒ x > 0 

⇒ x ∈ (0, ∞) 

Hence, Domain (f) = (0, ∞)