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

Options

A. A. `(-oo,0)`
B. B. `(-oo,oo)-{0}`
C. C. `(-oo,oo)`
D. D. `(0,oo)`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{|x| - x}} \), we need to ensure that the expression under the square root is positive and that the denominator is not zero.

### Step 1: Analyze the expression under the square root
The expression is \( |x| - x \). We need to determine when this expression is greater than zero.

### Step 2: Consider cases for \( |x| \)
The absolute value function \( |x| \) behaves differently based on whether \( x \) is positive or negative. We will consider two cases:

1. **Case 1: \( x \geq 0 \)**
- Here, \( |x| = x \).
- So, \( |x| - x = x - x = 0 \).
- Since we need \( |x| - x > 0 \), this case does not contribute to the domain.

2. **Case 2: \( x < 0 \)**
- Here, \( |x| = -x \).
- So, \( |x| - x = -x - x = -2x \).
- We need \( -2x > 0 \).
- Dividing both sides by -2 (and flipping the inequality), we get \( x < 0 \).

### Step 3: Determine the domain
From Case 2, we find that \( x \) must be less than 0 for the expression \( |x| - x \) to be positive. Therefore, the domain of the function is:

\[ (-\infty, 0) \]

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

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