of course, the two sides are equal.
the point is, if we define:
then , as you observed, but the "real interesting" part is:
so g is even, and:
so h is odd.
I am going to try to traduce the problem in english. :/
Demonstrate that any function , in interval , can be represented as the sum of even and odd functions.
Solution is: +
Why is that solution?
We just need to show that ?
Comment #1:
You have a mistake in your solution, not caused but translation into English; what is known as a "typo" -- a typographical error.
The solution should be:Comment #2.
The purpose of the restriction regarding the interval, is not at all clear to me. If by any chance the functions are supposed to be even or odd with respect to the center of the interval, then the functions, g(x) and h(x) mentioned by Deveno, should also include a coordinate translation.
Added in Edit:
The above equation is missing a plus sign. It should be:
I had a typo, correcting Fabio's typo is all.
Before Fabio edited his Original Post, he ha a couple of typos (See this quote in post #3). Then I inadvertently dropped the + sign between the two functions. I did a lot of cutting & pasting, because there are a lot of [TEX] [/TEX] tags in his equations.