Find the range of function
`(x+1)/(|x+2|)`

Answer

To find the range of the function \( f(x) = \frac{x+1}{|x+2|} \), we will analyze it in two cases based on the definition of the absolute value.

### Step 1: Analyze the function for \( x > -2 \)
When \( x > -2 \), the absolute value function simplifies to \( |x+2| = x+2 \). Therefore, the function becomes:
\[ f(x) = \frac{x+1}{x+2} \]

### Step 2: Set up the equation
Let \( f(x) = y \):
\[ y = \frac{x+1}{x+2} \]
Cross-multiplying gives:
\[ y(x + 2) = x + 1 \]
This simplifies to:
\[ yx + 2y = x + 1 \]

### Step 3: Rearranging the equation
Rearranging terms gives:
\[ yx - x = 1 - 2y \]
Factoring out \( x \) from the left-hand side:
\[ x(y - 1) = 1 - 2y \]
Thus, we can solve for \( x \):
\[ x = \frac{1 - 2y}{y - 1} \]

### Step 4: Determine the values of \( y \)
The function is defined for all \( y \) except where the denominator is zero. The denominator \( y - 1 \) is zero when \( y = 1 \). Therefore, for \( x > -2 \), the range is:
\[ \text{Range} = \mathbb{R} - \{1\} \]

### Step 5: Analyze the function for \( x < -2 \)
When \( x < -2 \), the absolute value function simplifies to \( |x+2| = -(x+2) \). Therefore, the function becomes:
\[ f(x) = \frac{x+1}{-(x+2)} = \frac{x+1}{-x-2} \]

### Step 6: Set up the equation
Let \( f(x) = y \):
\[ y = \frac{x+1}{-x-2} \]
Cross-multiplying gives:
\[ y(-x - 2) = x + 1 \]
This simplifies to:
\[ -yx - 2y = x + 1 \]

### Step 7: Rearranging the equation
Rearranging terms gives:
\[ -yx - x = 1 + 2y \]
Factoring out \( x \) from the left-hand side:
\[ x(-y - 1) = 1 + 2y \]
Thus, we can solve for \( x \):
\[ x = \frac{1 + 2y}{-y - 1} \]

### Step 8: Determine the values of \( y \)
The function is defined for all \( y \) except where the denominator is zero. The denominator \(-y - 1\) is zero when \( y = -1 \). Therefore, for \( x < -2 \), the range is:
\[ \text{Range} = \mathbb{R} - \{-1\} \]

### Step 9: Combine the ranges
Combining the ranges from both cases, we find:
\[ \text{Overall Range} = \mathbb{R} - \{1, -1\} \]

### Final Answer
The range of the function \( f(x) = \frac{x+1}{|x+2|} \) is:
\[ \text{Range} = \mathbb{R} - \{1, -1\} \]