Proving the Existence of Even and Odd Functions in f:R-->R

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The discussion revolves around proving that any function f: R → R can be expressed as the sum of an even function p and an odd function i. Participants suggest examining the relationship between f(-x) and the proposed functions p(x) and i(x). It is noted that if the theorem holds, then f(-x) can be represented as p(x) - i(x). The conversation highlights a breakthrough in understanding the relationship between even and odd functions. Ultimately, the participants acknowledge the insight gained from this exploration.
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I'm puzzled by this question: Show that for all function f:R-->R. there exists an even function p and an odd function i such that f(x) = p(x) + i(x) forall x in R.

I got nothing.
 
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Try looking at f(-x) and relating it to p(x) and i(x). Do you notice anything?
 
But there is nothing to look at. f(-x) = ......?

The only thing would be SUPPOSING the result of the thorem is true, then it would implies that there exist p and i such that f(x) = p+i and hence f(-x) = p(x)-i(x), but that's as far as that goes.
 
f(x) = p+i and hence f(-x) = p(x)-i(x), but that's as far as that goes.

No it's not.
 
Oh I see. That was very insightful Hurkyl. :-p
 

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