The domain of `y=(1)/(sqrt(|x|-x))` is :

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the conditions for the square root**:
The expression inside the square root must be greater than or equal to zero:
\[ |x| - x \geq 0 \]

2. **Analyze the absolute value**:
The absolute value function \( |x| \) behaves differently depending on whether \( x \) is non-negative or negative:
- If \( x \geq 0 \), then \( |x| = x \).
- If \( x < 0 \), then \( |x| = -x \).

3. **Case 1: \( x \geq 0 \)**:
If \( x \geq 0 \):
\[ |x| - x = x - x = 0 \]
This does not satisfy \( |x| - x > 0 \) (it equals zero), so we discard this case.

4. **Case 2: \( x < 0 \)**:
If \( x < 0 \):
\[ |x| - x = -x - x = -2x \]
We need to find when this is greater than zero:
\[ -2x > 0 \implies x < 0 \]
This condition is satisfied for all \( x < 0 \).

5. **Check the condition for the denominator**:
The denominator must not be equal to zero:
\[ |x| - x \neq 0 \]
From our previous analysis, we see that \( |x| - x = 0 \) only when \( x = 0 \) (which we already excluded) and is not an issue for \( x < 0 \).

6. **Conclusion**:
The only values of \( x \) that satisfy both conditions (the expression inside the square root being greater than zero and the denominator not being zero) are:
\[ x < 0 \]
Therefore, the domain of the function is:
\[ (-\infty, 0) \]

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