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 \( f(x) = \frac{|x|}{x} \) for \( x \neq 0 \), we will analyze the function based on the values of \( x \).

### Step 1: Define the function based on the sign of \( x \)

The absolute value function \( |x| \) behaves differently depending on whether \( x \) is positive or negative.

- If \( x > 0 \):
\[ |x| = x \implies f(x) = \frac{x}{x} = 1 \]

- If \( x < 0 \):
\[ |x| = -x \implies f(x) = \frac{-x}{x} = -1 \]

### Step 2: Determine the values of \( f(x) \)

From the analysis above, we can conclude:

- For all positive \( x \), \( f(x) = 1 \).
- For all negative \( x \), \( f(x) = -1 \).

### Step 3: Identify the range of the function

Since \( f(x) \) only takes the values 1 and -1 depending on whether \( x \) is positive or negative, we can summarize the range of \( f(x) \) as:

\[ \text{Range of } f(x) = \{-1, 1\} \]

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

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