Question

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

• A. \([0,\infty )\)
• B. \((0,\infty )\)
• C. \((-\infty ,0)\)
• D. \((-\infty ,0]\)

Hint:

The expression under the root must be strictly positive, so test |x| - x for positive and negative x.

Answer 1

The Correct Option is C

Step 1: Rewrite
$|x|-x$ equals $0$ for $x\ge 0$ and $-2x$ for $x<0$.

Step 2: Apply the condition
The denominator $\sqrt{|x|-x}$ must be nonzero and real, so $|x|-x>0$.

Step 3: Result
This holds only for $x<0$, so the domain is $(-\infty ,0)$.

Final Answer:
The function is defined only for x less than 0. \[ \boxed{\text{(C) }(-\infty ,0)} \]