Thread: Symmetric domain means the function is a sum of even and odd?

1. Symmetric domain means the function is a sum of even and odd?

Hello! I have an exercise that's bugging the out of me:

It's from Robert Adam's Calculus, sixth edition. Chapter P5, exercise 35. It goes like this:

"Let f be a function whose domain is symmetric about the origin, that is, -x belongs to the domain whenever x does. Show that f is the sum of an even function and an odd function

f(x) = E(x) + O(x)

where E is an even function and O is an odd function"

Then, in the second part of the exercise, we have "Show that there is only one way to write f as the sum of an even and an odd function".

Now, I don't have any issues with the calculation of the proof. Let E(x) = (f(x) + f(-x))/2 and go from there. It's the conclusion as such that I just can't get to fit in my head! I mean, take for example f(x) = x^2. By convention it's domain is the entire real line, so in other word it has a domain that's symmetrical about the origin, right? But just how is this supposed to work?

f(x) = E(x) + O(x)

I could take

f(x) = x^2 + 0x^3,

or I could go with

f(x) = x^2 + 0x^5.

Nothing unique about it. Really, what is E(x) and O(x) supposed to be in any given specific example of f?

So the conclusions of this exercise just confuses me. Could anyone maybe give me a hand in understanding how a symmetrical domain automatically gives that the function has to be a sum of an even and an odd dito? And thus help me untangle where I go wrong in my thinking with the example of f(x) = x^2?

2. Re: Symmetric domain means the function is a sum of even and odd?

Your $O(x)$ is identically zero in both cases - as you can tell from your proof where $O(x) = \frac{f(x)-f(-x)}2$.

3. Re: Symmetric domain means the function is a sum of even and odd?

Thanks for the reply!

Fair enough, but to be honest I'm still not sure I buy that. I mean, is the zero function, g(x) = 0, actually an odd function? My instinct would be to consider it even. I mean, 0 is an even number, and any constant function could be written as h(x) = Cx^0, right?

4. Re: Symmetric domain means the function is a sum of even and odd?

the function $f(x)=0$ with the given domain $D$ is both even and odd

it can be shown that it is the only such function

and that's what you need to show uniqueness:

If $f(x)=g(x)+h(x)=k(x)+m(x)$ where $g,k$ are even and $h,m$ are odd then

$g=k$ and $h=m$