1. **Stating the problem:** We are given the function $$y=\pi x \frac{x^2+1}{x^2-1}$$ and need to understand or analyze it. 2. **Formula and rules:** The function is a rational function multiplied by $$\pi x$$. Important rules: - The denominator $$x^2-1$$ cannot be zero, so $$x \neq \pm 1$$. - Simplify the expression if possible. 3. **Intermediate work:** The function is $$y=\pi x \frac{x^2+1}{x^2-1}$$. Note that $$x^2-1 = (x-1)(x+1)$$. No common factors with numerator, so no simplification. 4. **Explanation:** - The function is undefined at $$x=1$$ and $$x=-1$$ because the denominator is zero. - For other values of $$x$$, the function outputs $$\pi x$$ times the ratio of $$x^2+1$$ to $$x^2-1$$. 5. **Final answer:** The function is $$y=\pi x \frac{x^2+1}{x^2-1}$$ with domain $$x \in \mathbb{R} \setminus \{ -1,1 \}$$.