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 $x$.