Question: determine whether the following functions are even, odd, or neither: h(x) = $\frac{x}{x^2-1}$
1. **State the problem:** Determine if the function $$h(x) = \frac{x}{x^2 - 1}$$ is even, odd, or neither.
2. **Recall definitions:**
- A function $$f(x)$$ is **even** if $$f(-x) = f(x)$$ for all $$x$$ in the domain.
- A function $$f(x)$$ is **odd** if $$f(-x) = -f(x)$$ for all $$x$$ in the domain.
3. **Calculate $$h(-x)$$:**
$$h(-x) = \frac{-x}{(-x)^2 - 1} = \frac{-x}{x^2 - 1}$$
4. **Compare $$h(-x)$$ with $$h(x)$$:**
$$h(x) = \frac{x}{x^2 - 1}$$
$$h(-x) = \frac{-x}{x^2 - 1} = - \frac{x}{x^2 - 1} = -h(x)$$
5. **Conclusion:** Since $$h(-x) = -h(x)$$, the function $$h(x)$$ is **odd**.
**Final answer:** $$h(x) = \frac{x}{x^2 - 1}$$ is an odd function.