Question

Determine algebraically whether the function is even, odd, or neither even nor odd.

f(x) = -4x3 + 4x

Answer 1

For a function (fn) to be odd:
f(x) = - f(-x)
For a fn to be even:
f(x) = f(-x)
For a fn to be neither even nor odd
f(x) != f(-x) [No Relation]

(-x)^n = x^n for n -> even
(-x)^n = -x^n for n -> odd
In your example:

f(x) = -4x^3 + 4x
f(-x) = -4 (-x)^3 + 4 (-x)^1 ( 3 and 1 are odd powers )
f(-x) = 4x^3 - 4x (take -1 common to do the check)
f(-x) = -( -4x^3 + 4x ) = - f(x) [between the bracket was the original fn]

f(x) = - f(-x)
so the function is odd also called symmetric about the origin