To find the range of the function \( f(x) = \frac{x+2}{x^2 - 1} \), we need to analyze the behavior of the function and identify all the possible values that \( f(x) \) can take.

### Step-by-Step Process:

1. **Identify the domain:**
   First, find the values of \( x \) that make the function undefined. Since the denominator \( x^2 - 1 = (x-1)(x+1) \), the function is undefined when the denominator is zero. This happens at:
   \[
   x = 1 \quad \text{and} \quad x = -1
   \]
   Therefore, the function is undefined at \( x = 1 \) and \( x = -1 \), and these values are excluded from the domain.

2. **Examine asymptotic behavior:**
   - **Vertical Asymptotes:** The function has vertical asymptotes at \( x = 1 \) and \( x = -1 \), where the function goes to infinity or negative infinity.
   
   - **Horizontal Asymptote:** To find the horizontal asymptote, analyze the behavior of the function as \( x \to \infty \) or \( x \to -\infty \). For large values of \( x \), the highest degree terms dominate:
   \[
   f(x) \approx \frac{x}{x^2} = \frac{1}{x} \quad \text{as} \quad x \to \infty \quad \text{or} \quad x \to -\infty
   \]
   Therefore, \( f(x) \to 0 \) as \( x \to \infty \) or \( x \to -\infty \), so the horizontal asymptote is \( y = 0 \).

3. **Find the behavior near asymptotes:**
   - As \( x \to 1^+ \) or \( x \to 1^- \), the function tends to infinity or negative infinity.
   - As \( x \to -1^+ \) or \( x \to -1^- \), the function also tends to infinity or negative infinity.

4. **Critical points and checking values:**
   - The function might take all real values except 0 because \( y = 0 \) is a horizontal asymptote and cannot be reached. We should check if \( f(x) = 0 \) for any \( x \). Setting the numerator equal to zero:
   \[
   x + 2 = 0 \quad \Rightarrow \quad x = -2
   \]
   So, \( f(-2) = 0 \).

5. **Conclusion:**
   The function can take all real values except for the value it approaches as \( x \to \pm \infty \), which is 0. Thus, the range of the function is:
   \[
   \boxed{(-\infty, 0) \cup (0, \infty)}
   \]

Would you like further details or have any questions?

Here are some related questions:
1. What is the domain of \( f(x) = \frac{x+2}{x^2-1} \)?
2. How do you find the horizontal asymptote of a rational function?
3. Can a rational function cross its horizontal asymptote?
4. How does one find vertical asymptotes of a rational function?
5. What is the difference between vertical and horizontal asymptotes?

**Tip:** When analyzing rational functions, always check for both horizontal and vertical asymptotes to better understand the function's behavior at large and small values of \( x \).

Discover how to find the range of the rational function f(x) = (x+2)/(x^2-1). This problem explores key concepts like vertical and horizontal asymptotes and analyzes the behavior of the function. Suitable for Grades 10-12 math students.

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