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

f (x) = - 4x3 + 4x

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