Final answer:
The assertion that an even-degree polynomial function has opposite end behaviors is false, because even-degree polynomials actually have the same end behaviors—both ends either go towards positive infinity or negative infinity.
Step-by-step explanation:
The statement that an even-degree polynomial function has opposite end behaviors is false. For polynomial functions, the end behavior refers to what happens to the value of the function as x goes to positive or negative infinity. Polynomial functions with an even degree have the same end behavior on both ends, meaning they either both rise to positive infinity or both fall to negative infinity, depending on the leading coefficient.
An even function, defined by the property y(x) = y(-x), is symmetric about the y-axis, and this symmetry does not imply opposite end behaviors; rather, it reinforces similar behaviors for large positive and negative values of x. Contrarily, an odd function has opposite end behaviors, as it satisfies the property y(x) = −y(−x), leading to symmetry about the origin and making the function follow a pattern where one side of the x-axis is the mirror image of the other, inverted through the origin.