Expert: Jeff - 9/1/2004

Question
Hi...
Is there any way that you can explain this even and odd funtion deal to me? My professor did not explain it very well and I am having a hard time understanding it. If a polynomial has all even exponents does that make it an even function? And if it has all odd exponents does that make it odd? If so why? I need to see the reasoning or "proof" behind all of this so that I can better understand. Also, what happens if you have odd and even exponents in the same polynomial? Thanks in advance for any help that you may give.

Answer
An even function has the property that f(x)=f(-x)

so, f(x) = x^2 is even because f(-x) = (-x)^2 = (-1)^2 * x^2 = x^2 = f(x)

Similarily f(x) = x^(2n) = (x^n)^2

So any polynomial with only even exponents can be shown to be even.

An odd function is one where f(-x)=-f(x)

So, f(x) = x^3 is odd because

f(-x) = (-x)^3 = (-1)^3 * x^3 = -x^3 = -f(x)

If you have even AND odd exponents mixed together, chances are your polynomial is NEITHER even nor odd.

I hope this helps!

If you need online math tutoring or detailed solutions to math problems try visiting www.yourmathguide.com

Jeff