How do you determine if #x^3-x^2-x+1# is an even or odd function?

1 Answer
George C.
Apr 6, 2016

This function is neither even nor odd.

Explanation:

#f(x)# is an even function if #f(-x) = f(x)# for all #x#.

#f(x)# is an odd function if #f(-x) = -f(x)# for all #x#.

Let #f(x) = x^3-x^2-x+1#

Then:

#f(2) = 8-4-2+1 = 3#

#f(-2) = -8+4+2+1 = -1#

So #f(-2) != f(2)# and #f(-2) != -f(2)#

So this #f(x)# is neither even nor odd.

#color(white)()#
Actually, since this is a polynomial, we could tell by looking at the degrees of the terms:

A polynomial is an even function if and only if all of its terms are of even degree. Note that a constant term is of even degree since it has degree #0#.

A polynomial is an odd function if and only if all of its terms are of odd degree.

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The same is true of power series:

A power series represents an even function if and only if all of its terms are of even degree.

For example, #cos x = sum_(n=0)^oo (-1)^n x^(2n)/((2n)!)# is an even function.

A power series represents an odd function if and only if all of its terms are of odd degree.

For example, #sin x = sum_(n=0)^oo (-1)^n x^(2n+1)/((2n+1)!)# is an odd function.

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