Question

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

Answer 1

It's neither odd nor even since it has a non-zero `x^3` term (odd) and a non-zero constant term (even).

Explanation:

The quickest way to determine whether a polynomial is an odd or even function is to see if it has all odd degree terms or all even degree terms or a mixture.

In the case of this particular `f(x)`, the coefficient of the `x^3` term is `-2 * 1 * 1 * 4 = -8` and the constant term is `-2 * -3 * 2 * -3 = -36`

`x^3` is of degree `3` (odd) and the constant term is of degree `0` (even). So this polynomial is neither odd nor even.

To see directly, let's look at `f(+-1)`

`f(1) = -2 * -2 * 3 * 1 = 12`

`f(-1) = -2 * -4 * 1 * -7 = -56`

So neither `f(-1) = f(1)` nor `f(-1) = -f(1)`

Even functions satisfy `f(-x) = f(x)` for all `x`.

Odd functions satisfy `f(-x) = -f(x)` for all `x`.

So `f(x)` is neither.