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

1 Answer
George C.
Oct 18, 2015

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

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