The domain of the function `f(x)=(x)/(|x|)` is :

Options

A. `R-{0}`
B. R
C. Z
D. W

Answer

To find the domain of the function \( f(x) = \frac{x}{|x|} \), we need to determine the values of \( x \) for which this function is defined.

### Step-by-step Solution:

1. **Understanding the Function**:
The function \( f(x) = \frac{x}{|x|} \) involves the absolute value of \( x \). The absolute value function \( |x| \) is defined for all real numbers, but we need to consider how it affects our function.

2. **Identifying the Absolute Value**:
The absolute value \( |x| \) is defined as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

3. **Finding the Function's Behavior**:
- For \( x > 0 \):
\[ f(x) = \frac{x}{|x|} = \frac{x}{x} = 1 \]
- For \( x < 0 \):
\[ f(x) = \frac{x}{|x|} = \frac{x}{-x} = -1 \]
- For \( x = 0 \):
The function becomes \( f(0) = \frac{0}{|0|} \), which is undefined because we cannot divide by zero.

4. **Conclusion on Domain**:
Since the function is defined for all real numbers except \( x = 0 \), we conclude that the domain of \( f(x) \) is:
\[ \text{Domain} = \mathbb{R} \setminus \{0\} \]
or in interval notation:
\[ (-\infty, 0) \cup (0, \infty) \]

### Final Answer:
The domain of the function \( f(x) = \frac{x}{|x|} \) is all real numbers except 0.