Rational Function Analysis

The animation visualizes the behavior of the function (f(x)=\dfrac{x(1-x)^2}{(1+x^2)^2}) by demonstrating its reciprocal symmetry, showing that it is not one-to-one, and illustrating how its maximum value is bounded by a sum-of-squares inequality. It concludes by using a substitution to reveal that the function's exact range of values is ([-1, 0] \cup (0, 1/8