What is the domain of the function `f(x)=(1)/(sqrt(|x|-x))`?

Options

A. `(-oo,0)`
B. `(0,oo)`
C. `0ltxlt1`
D. `xgt1`

Explanation

To find the domain of the function \( f(x) = \frac{1}{\sqrt{|x| - x}} \), we need to ensure that the expression inside the square root is positive, as the square root function is only defined for non-negative values, and we cannot have a zero in the denominator.

### Step 1: Analyze the expression inside the square root
The expression inside the square root is \( |x| - x \). We need to analyze this based on the definition of the absolute value.

### Step 2: Consider cases for \( |x| \)
1. **Case 1: \( x \geq 0 \)**
- Here, \( |x| = x \).
- Thus, \( |x| - x = x - x = 0 \).
- Since the square root of zero is not positive, this case does not contribute to the domain.

2. **Case 2: \( x < 0 \)**
- In this case, \( |x| = -x \).
- Therefore, \( |x| - x = -x - x = -2x \).
- We need \( -2x > 0 \) for the square root to be defined.
- Solving this inequality:
\[ -2x > 0 \implies x < 0 \]
- This condition is satisfied for all \( x < 0 \).

### Step 3: Conclusion about the domain
From our analysis, the only values of \( x \) that keep the expression \( |x| - x \) positive are those where \( x < 0 \). Therefore, the domain of the function is:

\[ \text{Domain: } (-\infty, 0) \]

### Final Answer
The domain of the function \( f(x) = \frac{1}{\sqrt{|x| - x}} \) is \( (-\infty, 0) \).

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