What is the range of the function `f(x)=(|x|)/(x), xne0`?

Options

A. Set of all real numbers
B. Set of all integers
C. {-1, 1}
D. {-1, 0, 1}

Explanation

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

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

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

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

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

### Step 2: Determine the outputs for both cases

From the analysis above, we can see that:

- For \( x > 0 \), \( f(x) = 1 \)
- For \( x < 0 \), \( f(x) = -1 \)

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

Since the function \( f(x) \) only takes the values \( 1 \) and \( -1 \), we can conclude that the range of the function is:

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

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

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