1. The problem is to analyze the function given by $$x = \frac{1 + t^2}{1 - t^2}$$. 2. This is a rational function where the numerator is $$1 + t^2$$ and the denominator is $$1 - t^2$$. 3. Note that the function is undefined where the denominator is zero, i.e. when $$1 - t^2 = 0$$, or $$t^2 = 1$$ which implies $$t = \pm 1$$. 4. For values of $$t$$ such that $$|t| < 1$$, the denominator is positive and the numerator is always positive or zero. 5. For values where $$|t| > 1$$, the denominator is negative while the numerator remains positive, so the function may become negative. 6. Simplify where possible: the expression cannot be simplified further algebraically. 7. The domain of $$x$$ is all real numbers $$t$$ except $$\pm 1$$. 8. Important behavior: - Vertical asymptotes at $$t = \pm 1$$. - When $$t = 0$$, $$x = \frac{1+0}{1-0} = 1$$. - As $$t \to \pm \infty$$, $$t^2 \to +\infty$$, so $$x \approx \frac{t^2}{-t^2} = -1$$. Final answer: The function is $$x = \frac{1 + t^2}{1 - t^2}$$ with domain $$t \neq \pm 1$$, vertical asymptotes at $$t = \pm 1$$, and horizontal asymptote $$x = -1$$ as $$t \to \pm \infty$$.