Question

How do you determine if `f(x)=x^5-x+5` is an even or odd function?

Answer 1

`f(x)` is neither even nor odd.

Explanation:

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

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

For polynomials, there is a quick test of whether it is odd or even:

If all of the terms are of odd degree then it is odd.

If all of the terms are of even degree then it is even.

Otherwise it is neither odd nor even.

In our example, `x^5` and `-x` are of odd degree and the constant `5` is of even (`0`) degree. So `f(x)` is neither odd nor even.

Specifically we find:

`f(2) = 32-2+5 = 35`

`f(-2) = -32+2+5 = -25`

So `f(-2)` is neither `f(2)` nor `-f(2)`.