The domain of `f(x) = (4)/(|x|-x)` is

Options

A. `x lt - 4`
B. `x ge 0`
C. `x lt 0`
D. `x gt 1`

Explanation

To find the domain of the function \( f(x) = \frac{4}{|x| - x} \), we need to determine the values of \( x \) for which the function is defined. Since the function has a denominator, we must ensure that the denominator is not equal to zero.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is \( |x| - x \).

2. **Set the Denominator Not Equal to Zero**:
We need to solve the inequality:
\[ |x| - x \neq 0 \]

3. **Consider Cases for Absolute Value**:
The expression \( |x| \) behaves differently based on whether \( x \) is non-negative or negative. We will consider two cases:
- **Case 1**: \( x \geq 0 \)
- **Case 2**: \( x < 0 \)

4. **Case 1: \( x \geq 0 \)**:
In this case, \( |x| = x \). Therefore:
\[ |x| - x = x - x = 0 \]
This means that when \( x \geq 0 \), the denominator is zero. Thus, \( f(x) \) is undefined for \( x \geq 0 \).

5. **Case 2: \( x < 0 \)**:
Here, \( |x| = -x \). Therefore:
\[ |x| - x = -x - x = -2x \]
We need to check when this is not equal to zero:
\[ -2x \neq 0 \implies x \neq 0 \]
Since we are already in the case where \( x < 0 \), this condition is satisfied for all negative values of \( x \).

6. **Conclusion**:
The function \( f(x) \) is defined for all values of \( x \) that are negative. Therefore, the domain of \( f(x) \) is:
\[ \text{Domain} = (-\infty, 0) \]

### Final Answer:
The domain of \( f(x) = \frac{4}{|x| - x} \) is \( x < 0 \).