Question

How do you determine if `f(x) = 3x^4 - x^2 + 2` is an even or odd function?

Answer 1

This `f(x)` is an even function.

Explanation:

• An even function is one where `f(-x) = f(x)` for all `x` in the domain.

• An odd function is one where `f(-x) = -f(x)` for all `x` in the domain.

In the case of `f(x) = 3x^4-x^2+2` we find:

`f(-x) = 3(-x)^4-(-x)^2+2 = 3x^4-x^2+2 = f(x)`

So `f(x)` is an even function.

Actually, there is a shortcut with polynomial functions:

• If all of the terms have even degree then the function is even.

• If all of the terms have odd degree then the function is odd.

• Otherwise the function is neither odd nor even.

In our example, `color(blue)(3x^4)` has degree `4`, `color(blue)(-x^2)` has degree `2` and `color(blue)(2)` has degree `0`. So all of the terms are of even degree and the function is even.