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

Options

A. `[0,oo)`
B. `(-oo,0)`
C. `[1,oo)`
D. `(-oo,0]`

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 and that the denominator is not equal to zero. Let's go through the steps to determine the domain.

### Step 1: Identify the expression under the square root
The expression we need to analyze is \( |x| - x \).

### Step 2: Analyze the cases for \( |x| \)
The absolute value function \( |x| \) behaves differently based on whether \( x \) is positive or negative:
- If \( x \geq 0 \), then \( |x| = x \).
- If \( x < 0 \), then \( |x| = -x \).

### Step 3: Case 1 - When \( x \geq 0 \)
If \( x \geq 0 \):
\[ |x| - x = x - x = 0 \]
In this case, the expression under the square root becomes zero, which means the function is undefined (since we cannot divide by zero). Therefore, \( x \geq 0 \) is not in the domain.

### Step 4: Case 2 - When \( x < 0 \)
If \( x < 0 \):
\[ |x| - x = -x - x = -2x \]
For the square root to be defined and positive:
\[ -2x > 0 \implies x < 0 \]
This condition is satisfied for all \( x < 0 \).

### Step 5: Conclusion
Since the function is undefined for \( x \geq 0 \) and valid for \( x < 0 \), the domain of the function is:
\[ (-\infty, 0) \]

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

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