Find the domain and range of `f(x)=(x+2)/(|x+2|)`.

Answer

To find the domain and range of the function \( f(x) = \frac{x+2}{|x+2|} \), we will follow these steps:

### Step 1: Determine when the function is undefined
The function \( f(x) \) is undefined when the denominator is zero. Therefore, we need to solve the equation:
\[ |x + 2| = 0 \]
This occurs when:
\[ x + 2 = 0 \implies x = -2 \]

### Step 2: State the domain
Since \( f(x) \) is undefined at \( x = -2 \), the domain of \( f(x) \) includes all real numbers except \( -2 \). We can express the domain in interval notation:
\[ \text{Domain} = (-\infty, -2) \cup (-2, \infty) \]

### Step 3: Analyze the function for the range
Next, we will analyze the function to determine its range. The function can be simplified based on the value of \( x + 2 \):
- If \( x + 2 > 0 \) (i.e., \( x > -2 \)), then \( |x + 2| = x + 2 \) and:
\[ f(x) = \frac{x + 2}{x + 2} = 1 \]
- If \( x + 2 < 0 \) (i.e., \( x < -2 \)), then \( |x + 2| = -(x + 2) \) and:
\[ f(x) = \frac{x + 2}{-(x + 2)} = -1 \]

### Step 4: State the range
From the analysis above, we see that:
- For \( x > -2 \), \( f(x) = 1 \)
- For \( x < -2 \), \( f(x) = -1 \)

Thus, the range of \( f(x) \) consists of the two values:
\[ \text{Range} = \{-1, 1\} \]

### Final Answer
- **Domain**: \( (-\infty, -2) \cup (-2, \infty) \)
- **Range**: \( \{-1, 1\} \)