`A = { x // x in R, x != 0, -4 <= x <= 4` and `f: A -> R` is defined by `f(x) = |x| / x` for `x in A`. Then the range of f is

Options

A. {1,-1}
B. `{x:0 le x le 4}`
C. {1}
D. `{x: -4 le x le 0}`

Answer

To find the range of the function \( f: A \to \mathbb{R} \) defined by \( f(x) = \frac{|x|}{x} \) for \( x \in A \), where \( A = \{ x \in \mathbb{R} \mid x \neq 0, -4 \leq x \leq 4 \} \), we can follow these steps:

### Step 1: Identify the domain
The set \( A \) consists of all real numbers \( x \) such that \( -4 \leq x \leq 4 \) and \( x \neq 0 \). Therefore, we can express \( A \) as:
\[ A = [-4, 0) \cup (0, 4] \]

### Step 2: Analyze the function \( f(x) \)
The function \( f(x) = \frac{|x|}{x} \) can be analyzed based on the sign of \( x \):
- If \( x > 0 \), then \( |x| = x \), so:
\[ f(x) = \frac{x}{x} = 1 \]
- If \( x < 0 \), then \( |x| = -x \), so:
\[ f(x) = \frac{-x}{x} = -1 \]

### Step 3: Determine the output values for the intervals in \( A \)
- For \( x \in (0, 4] \) (positive values), \( f(x) = 1 \).
- For \( x \in [-4, 0) \) (negative values), \( f(x) = -1 \).

### Step 4: Conclusion about the range
Since \( f(x) \) takes the value \( 1 \) for all positive \( x \) in \( (0, 4] \) and the value \( -1 \) for all negative \( x \) in \( [-4, 0) \), the range of \( f \) is:
\[ \text{Range of } f = \{-1, 1\} \]

### Final Answer
The range of the function \( f \) is \( \{-1, 1\} \).

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