Question

How do you determine if `f(x)=8x^6 + 12x^2` is an even or odd function?

Answer 1

It's an even function.

Explanation:

One of the easy ways to determine weather a function is even or odd is to look at the powers of `x`. If all the powers of `x` are even, such as `8x^6+12x^2` (powers are `6` and `2`), then it's an even function. If all the powers of `x` are odd, such as `5x^3+` `x` (powers of `x` are `3` and `1`) , then it's an odd function.

Also remember that a function can be neither odd or even function, such as;
`f(x)=5x^3+x^4`

Answer 2

Verify `f(-x) = f(x)` for all `x in RR`, so `f(x)` is even.

Explanation:

`f(-x) = 8(-x)^6+12(-x)^2 = 8x^6+12x^2 = f(x)` for all `x in RR`

So `f(x)` is even.

For polynomial functions, there is a quick shortcut:

If all of the terms have even degree then the function is even. Remember that a constant term is of degree `0` which is even.

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

If the terms are a mixture of odd and even degrees then the function is neither even nor odd.