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

Options

A. `R-[3,-2]`
B. `R-[-3,2]`
C. `[R-[3,-2]`
D. `R-(-3-2)`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{|x| - x}} \), we need to determine the values of \( x \) for which the function is defined. This involves ensuring that the denominator is not zero and that the expression inside the square root is non-negative.

### Step-by-Step Solution:

1. **Identify the expression inside the square root**:
We have \( |x| - x \) in the denominator. The function \( f(x) \) is defined as long as \( \sqrt{|x| - x} \) is greater than 0.

2. **Consider the cases for \( |x| \)**:
The absolute value function \( |x| \) behaves differently depending on whether \( x \) is positive or negative. We will analyze two cases: when \( x \geq 0 \) and when \( x < 0 \).

3. **Case 1: \( x \geq 0 \)**:
- If \( x \geq 0 \), then \( |x| = x \).
- Therefore, \( |x| - x = x - x = 0 \).
- Since \( \sqrt{0} = 0 \), the denominator becomes zero, which means \( f(x) \) is not defined for \( x \geq 0 \).

4. **Case 2: \( x < 0 \)**:
- If \( x < 0 \), then \( |x| = -x \).
- Therefore, \( |x| - x = -x - x = -2x \).
- Since \( x < 0 \), \( -2x > 0 \) (because multiplying a negative number by -2 gives a positive result).
- Thus, \( \sqrt{-2x} \) is defined and positive for \( x < 0 \).

5. **Conclusion**:
- The function \( f(x) \) is not defined for \( x \geq 0 \) (including \( x = 0 \)).
- The function \( f(x) \) is defined for all \( x < 0 \).
- Therefore, the domain of \( f(x) \) is \( (-\infty, 0) \).

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