1. **State the problem:** Find the domain, vertical asymptotes, and horizontal asymptotes of the function $$f(x) = \frac{x}{x^2 - 1}$$. 2. **Domain:** The domain of a function includes all real numbers except where the denominator is zero because division by zero is undefined. 3. **Find values that make the denominator zero:** Solve $$x^2 - 1 = 0$$. 4. Factor the denominator: $$x^2 - 1 = (x - 1)(x + 1)$$. 5. Set each factor equal to zero: $$x - 1 = 0 \Rightarrow x = 1$$ and $$x + 1 = 0 \Rightarrow x = -1$$. 6. **Domain conclusion:** The domain is all real numbers except $$x = -1$$ and $$x = 1$$. 7. **Vertical asymptotes:** Vertical asymptotes occur where the denominator is zero and the numerator is not zero at those points. 8. Since numerator $$x$$ is not zero at $$x = \pm 1$$, vertical asymptotes are at $$x = -1$$ and $$x = 1$$. 9. **Horizontal asymptotes:** Compare degrees of numerator and denominator. 10. Degree of numerator is 1 (since $$x$$), degree of denominator is 2 (since $$x^2$$). 11. When degree of denominator > degree of numerator, horizontal asymptote is $$y = 0$$. 12. **Final answers:** - Domain: all real numbers except $$x = -1, 1$$ - Vertical asymptotes: $$x = -1, 1$$ - Horizontal asymptote: $$y = 0$$