The domain of the function f(x) = \(\rm \frac{1}{\sqrt{|x|-x}}\) is :

Concept:

Domain of a function:

• The domain of a function is the set of all input values (x-values) for which the function is defined.
• For a function containing a square root, the expression inside the root must be ≥ 0.
• If the root is in the denominator, then the expression must be > 0 (since division by zero is undefined).
• Here, the function is f(x) = 1 / √(|x| - x).
• So, we must ensure that |x| - x > 0 for f(x) to be defined.

Calculation:

Given,

f(x) = 1 / √(|x| - x)

We need: |x| - x > 0

⇒ Consider two cases for x:

⇒ Case 1: x ≥ 0 ⇒ |x| = x ⇒ |x| - x = x - x = 0 (Not allowed)

⇒ Case 2: x < 0 ⇒ |x| = -x ⇒ |x| - x = -x - x = -2x > 0

⇒ This is true for all x < 0

∴ Domain of the function is (-∞, 0)