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

Question

Theodore Amidon · Accepted Answer

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

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.

Serenity Johnson · Answer

undefined To determine if the function $ f(x) = -2(x - 3)(x + 2)(4x - 3) $ is even or odd:

1. Substitute $ -x $ for $ x $ in the function.
2. Simplify the expression.
3. If the result is equal to the original function ($ f(-x) = f(x) $), the function is even.
4. If the result is equal to the negation of the original function ($ f(-x) = -f(x) $), the function is odd.

Performing the substitution and simplification:

$$
f(-x) = -2(-x - 3)(-x + 2)(4(-x) - 3)
$$

$$
= -2(-x - 3)(-x + 2)(-4x - 3)
$$

Comparing with the original function:

$$
f(-x) = -f(x)
$$

Therefore, the function $ f(x) = -2(x - 3)(x + 2)(4x - 3) $ is an odd function.