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

1 Answer
Oct 18, 2015

Answer:

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#