The range of the function `f(x)=(x+2)/(|x+2|),\ x!=-2`

Options

A. `{-1,1}`
B. `{-1,0,1}`
C. `{1}`
D. `(0,oo)`

Explanation

To find the range of the function \( f(x) = \frac{x + 2}{|x + 2|} \) where \( x \neq -2 \), we can analyze the function based on the properties of the absolute value.

### Step 1: Understand the Absolute Value
The absolute value function \( |x + 2| \) behaves differently based on the value of \( x + 2 \):
- If \( x + 2 \geq 0 \) (i.e., \( x \geq -2 \)), then \( |x + 2| = x + 2 \).
- If \( x + 2 < 0 \) (i.e., \( x < -2 \)), then \( |x + 2| = -(x + 2) = -x - 2 \).

### Step 2: Define the Function in Two Cases
We will split the function into two cases based on the value of \( x \):

1. **Case 1:** \( x > -2 \)
- Here, \( |x + 2| = x + 2 \).
- Thus, \( f(x) = \frac{x + 2}{x + 2} = 1 \) (for \( x > -2 \)).

2. **Case 2:** \( x < -2 \)
- Here, \( |x + 2| = - (x + 2) = -x - 2 \).
- Thus, \( f(x) = \frac{x + 2}{-x - 2} = -1 \) (for \( x < -2 \)).

### Step 3: Determine the Range
From the two cases:
- For \( x > -2 \), \( f(x) = 1 \).
- For \( x < -2 \), \( f(x) = -1 \).

Since \( x \neq -2 \), the function does not take any other values. Therefore, the range of the function is:
\[ \text{Range} = \{1, -1\} \]

### Conclusion
The range of the function \( f(x) = \frac{x + 2}{|x + 2|} \) where \( x \neq -2 \) is \( \{-1, 1\} \).

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