The domain of the function `f(x)=1/(sqrt(|x|-x))` is: (1) `(-oo,oo)` (2) `(0,oo` (3) `(-oo,""0)` (4) `(-oo,oo)"-"{0}`

Options

A. `(-oo,oo) ~{0}`
B. `(-oo,oo)`
C. `(0,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 non-negative and that the denominator is not equal to zero. Let's break this down step by step.

### Step 1: Analyze the expression inside the square root

The expression inside the square root is \( |x| - x \). We need to determine when this expression is non-negative:

1. **Case 1:** When \( x \geq 0 \)
- Here, \( |x| = x \).
- Thus, \( |x| - x = x - x = 0 \).
- The square root \( \sqrt{0} \) is defined, but the denominator becomes zero, which makes the function undefined.

2. **Case 2:** When \( x < 0 \)
- Here, \( |x| = -x \).
- Thus, \( |x| - x = -x - x = -2x \).
- We need \( -2x \geq 0 \) for the square root to be defined.
- This simplifies to \( x \leq 0 \).

### Step 2: Determine the conditions for the function to be defined

From the analysis:
- For \( x \geq 0 \), the function is undefined because the denominator becomes zero.
- For \( x < 0 \), the expression \( -2x \) is non-negative, which means the square root is defined.

### Step 3: Conclusion about the domain

The function \( f(x) \) is defined for all negative values of \( x \). Therefore, the domain of the function is:

\[ (-\infty, 0) \]

### Final Answer

The correct option is:
(3) \( (-\infty, 0) \)

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