Odd and Even Function Constructions

Date: 06/08/98 at 20:35:00
From: C. Brinson
Subject: Odd and even function constructions

If f is an odd function and g is an even function, how do we go about 
combining the two in various constructions (such as adding the two, 
multiplying the two, composing one with the other, etc.)? How do we 
know if the "new" function is even or odd?  I know there is an easy 
way to figure this out, like substituting x with (-x), but I'm stuck 
and it's been a while since I've had to think about such things.  

Thanks.  
C. Brinson

Date: 06/16/98 at 21:50:14
From: Doctor Schwa
Subject: Re: Odd and even function constructions

An odd function is one where f(-x) = -f(x), and an even function is 
one where g(-x) = g(x). That is, x^(odd power) is an example of an 
odd function, and x^(even power) is an example of an even function.

So for example, if you multiply an odd and an even function, you 
can tell from the x^(power) example that the result will be odd, 
because the powers add. You can prove it in general by taking
fg(-x) = f(-x)g(-x) = -f(x) g(x) = -(fg(x))

When you add an odd and an even function, in general you'll get 
a function that's neither odd nor even. In fact any function h(x) 
can be written as a sum of an odd and even part, by taking
f(x) = (h(x) - h(-x))/2 as the odd part, and
g(x) = (h(x) + h(-x))/2 for the even part.

-Doctor Schwa,  The Math Forum
 http://mathforum.org/dr.math/