Question

Is the function `f(x)= -4x^2 + 4x` even, odd or neither?

Answer 1

Neither

Explanation:

The quick way to spot whether a polynomial function in `x` is odd or even is the powers of `x` that occur. If they are all odd then the polynomial is odd. If they are all even then the polynomial is even. Note that a constant is an even power of `x` - namely `x^0`.

By definition:

`f(x)` is odd if `f(-x) = -f(x)` for all `x` in the domain.

`f(x)` is even if `f(-x) = f(x)` for all `x` in the domain.

In our case, we find:

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

`f(1) = -4+4 = 0`

So neither condition holds.

Given any function `f(x)`, it can be expressed uniquely as the sum of an even function and an odd function, defined as follows:

`f_e(x) = (f(x) + f(-x))/2`

`f_o(x) = (f(x) - f(-x))/2`

In our case we find `f_e(x) = -4x^2` and `f_o(x) = 4x`