How do you know if #f(x) = -2(x - 3) (x + 2) (4x - 3)# is an even or odd function?

1 Answer
George C.
Nov 28, 2015

It's neither odd nor even since it has a non-zero #x^3# term (odd) and a non-zero constant term (even).

Explanation:

The quickest way to determine whether a polynomial is an odd or even function is to see if it has all odd degree terms or all even degree terms or a mixture.

In the case of this particular #f(x)#, the coefficient of the #x^3# term is #-2 * 1 * 1 * 4 = -8# and the constant term is #-2 * -3 * 2 * -3 = -36#

#x^3# is of degree #3# (odd) and the constant term is of degree #0# (even). So this polynomial is neither odd nor even.

To see directly, let's look at #f(+-1)#

#f(1) = -2 * -2 * 3 * 1 = 12#

#f(-1) = -2 * -4 * 1 * -7 = -56#

So neither #f(-1) = f(1)# nor #f(-1) = -f(1)#

Even functions satisfy #f(-x) = f(x)# for all #x#.

Odd functions satisfy #f(-x) = -f(x)# for all #x#.

So #f(x)# is neither.

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