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March 30th, 2014, 09:12 AM   #1
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Post function equation

Find all so that
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March 30th, 2014, 07:39 PM   #2
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Quote:
Originally Posted by gelatine1 View Post
Find all so that

or

?

The latter makes more sense to me on the face of it.
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March 30th, 2014, 07:54 PM   #3
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For the equation originally posted, f(x) ≡ 0 and f(x) ≡ -x work. I suspect they are the only solutions, but I don't have a proof of that.
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March 31st, 2014, 10:39 AM   #4
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Yes, that is correct.
Although I have a proof (in Dutch) I don't completely understand it. But at least your solutions are correct
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March 31st, 2014, 09:56 PM   #5
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Well, I can see those two solutions:

If we assume that there exists a value of y for which f(y) = 0, then we have that f(xy) = 0 for all x. This gives us two possibilities:
  1. f(x) = 0 for all x; or
  2. y = 0, which means that f(0) = 0.
The first is clearly degenerate, so let's take f(0) = 0. Note that we got this far on the basis that there exists at least one value for which f(y) = 0.

Moving on, if we let x = 0 in the functional equation, we get

because f(0) = 0.

Now, if we assume that there exists an inverse function g(x) such that g(f(x)) = x, we can apply this function to both sides of our equation to get

so
for all y.

Thus, if there exists a value of x for which f(x) = 0, we can say that



I can't make anything work if we assume that there is no value for which f(x) = 0. That is, I can neither see such a function nor prove that no such function exists.
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