Question

How do you determine if `f(x)=x^4 - x^12 +1` is an even or odd function?

Answer 1

`f(x)` is an even function since `f(-x) = f(x)` for all `x` in its domain.

Explanation:

• An even function is one that satisfies `f(-x) = f(x)` for all `x` in its domain.

• An odd function is one that satisfies `f(-x) = -f(x)` for all `x` in its domain.

In our example:

`f(-x) = (-x)^4-(-x)^12+1 = x^4-x^12+1 = f(x)`

for any value of `x` (the domain of `f` is the whole of `RR`).

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

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Footnote

There is a shortcut to determine whether a polynomial or rational function `f(x)` is even or odd:

• If `f(x)` only contains even powers of `x`, then it is an even function.

• If `f(x)` only contains odd powers of `x`, then it is an odd function.

• If `f(x)` contains a mixture of terms of odd and even degree then it is neither odd nor even.

Note that constant terms are of even degree `(0)`.