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Math Help - Function Problem

  1. #1
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    Question Function Problem

    Given f(x)=65 and f(x)f(1/x)=f(x)+f(1/x)... find the value of f(6)=???
    Last edited by mr fantastic; May 28th 2011 at 03:44 PM. Reason: Title.
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  2. #2
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    Quote Originally Posted by avik View Post
    Given f(x)=65 and f(x)f(1/x)=f(x)+f(1/x)... find the value of f(6)=???
    Surely that is a mistake (typo?) in what you have posted.
    If so, please correct it.
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    Yes I'm Sorry. That is f(4)=65.
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    There is not enough information here for the problem to have a unique solution. I suspect that the intended answer is that f(x) = x^3+1, and so f(6) = 217. But in fact if g(x) is an arbitrary function such that g(1/x) = g(x) and g(4)=3, then f(x) = x^{g(x)}+1 is a solution, and f(6) can be a completely arbitrary number greater than 1.
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    Quote Originally Posted by Opalg View Post
    There is not enough information here for the problem to have a unique solution. I suspect that the intended answer is that f(x) = x^3+1, and so f(6) = 217. But in fact if g(x) is an arbitrary function such that g(1/x) = g(x) and g(4)=3, then f(x) = x^{g(x)}+1 is a solution, and f(6) can be a completely arbitrary number greater than 1.
    This is what given in the problem. Anyways, that 217 answer is correct. Can you please explain how did you deduce f(x) = x^3+1 from the problem statement? please reply..
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  6. #6
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    Quote Originally Posted by avik View Post
    Can you please explain how did you deduce f(x) = x^3+1 from the problem statement? please reply..
    The equation f(x)f(1/x)=f(x)+f(1/x) can be written as \bigl((f(x)-1\bigr)\bigl((f(1/x)-1\bigr) = 1. So if g(x) = f(x)-1 then g(1/x) = 1/g(x). There are very many functions having that property, but the most obvious solutions are g(x) = x^k for some fixed k.

    If f(4) = 65 then g(4) = 64 = 4^3. That suggests that we should take k=3, giving g(x) = x^3 and hence f(x) = x^3+1.

    But as I said in the previous comment, there is nothing unique about that solution. If there really was no further information given, then this is a very badly thought out problem.
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