The range of f(x) = `(|x|)/(x), xne 0` is :

Options

A. [0,1]
B. [-1,1]
C. (-1,1)
D. {-1,1}

Answer

To find the range of the function \( f(x) = \frac{|x|}{x} \) where \( x \neq 0 \), we can analyze the function based on the values of \( x \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = \frac{|x|}{x} \) involves the absolute value of \( x \). We need to consider two cases based on the sign of \( x \).

2. **Case 1: When \( x > 0 \)**:
- In this case, \( |x| = x \).
- Therefore, \( f(x) = \frac{x}{x} = 1 \).
- So, for all positive values of \( x \), \( f(x) = 1 \).

3. **Case 2: When \( x < 0 \)**:
- Here, \( |x| = -x \).
- Thus, \( f(x) = \frac{-x}{x} = -1 \).
- So, for all negative values of \( x \), \( f(x) = -1 \).

4. **Conclusion**:
- From the two cases, we see that the function takes the value \( 1 \) when \( x > 0 \) and \( -1 \) when \( x < 0 \).
- Therefore, the range of \( f(x) \) is the set of values that \( f(x) \) can take, which is \( \{-1, 1\} \).

### Final Answer:
The range of \( f(x) = \frac{|x|}{x} \) where \( x \neq 0 \) is \( \{-1, 1\} \).