Question

How do you determine if `x^3-x^2-x+1` is an even or odd function?

Answer 1

This function is neither even nor odd.

Explanation:

`f(x)` is an even function if `f(-x) = f(x)` for all `x`.

`f(x)` is an odd function if `f(-x) = -f(x)` for all `x`.

Let `f(x) = x^3-x^2-x+1`

Then:

`f(2) = 8-4-2+1 = 3`

`f(-2) = -8+4+2+1 = -1`

So `f(-2) != f(2)` and `f(-2) != -f(2)`

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

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Actually, since this is a polynomial, we could tell by looking at the degrees of the terms:

A polynomial is an even function if and only if all of its terms are of even degree. Note that a constant term is of even degree since it has degree `0`.

A polynomial is an odd function if and only if all of its terms are of odd degree.

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The same is true of power series:

A power series represents an even function if and only if all of its terms are of even degree.

For example, `cos x = sum_(n=0)^oo (-1)^n x^(2n)/((2n)!)` is an even function.

A power series represents an odd function if and only if all of its terms are of odd degree.

For example, `sin x = sum_(n=0)^oo (-1)^n x^(2n+1)/((2n+1)!)` is an odd function.