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Math Help - Two questions about function composition

  1. #1
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    Two questions about function composition

    #1
    I have a continuous, one to one function f(x). How do I find g(x) such that g(g(x)) = f(x)?

    #2
    Also, if i forever keep feeding the output of a continuous one to one funtion back into the input, isn't the result also continuous and one to one?

    {i mean ...f(f(f(f(...f(x)))...}

    How do find the values of x for which this converges to some finite value?

    Can I find what the value is?

    thanks
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  2. #2
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    Quote Originally Posted by kd8bxz View Post
    #1
    I have a continuous, one to one function f(x). How do I find g(x) such that g(g(x)) = f(x)?

    #2
    Also, if i forever keep feeding the output of a continuous one to one funtion back into the input, isn't the result also continuous and one to one?

    {i mean ...f(f(f(f(...f(x)))...}

    How do find the values of x for which this converges to some finite value?

    Can I find what the value is?

    thanks
    Read this: Iterated function - Wikipedia, the free encyclopedia
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  3. #3
    Super Member Failure's Avatar
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    Quote Originally Posted by kd8bxz View Post
    #1
    I have a continuous, one to one function f(x). How do I find g(x) such that g(g(x)) = f(x)?
    Sorry, but such a function g (with the same domain and range as f) need not exist. Consider for example, the function f: M\ni x\mapsto -x\in M, where  M := \{-1;+1\}. In this example it is easy to have an overview of all the possible Functions g:M\rightarrow M. Do you see one that satisfies g(g(x))=f(x) for all x\in M?

    #2
    Also, if i forever keep feeding the output of a continuous one to one funtion back into the input, isn't the result also continuous and one to one?

    {i mean ...f(f(f(f(...f(x)))...}

    How do find the values of x for which this converges to some finite value?

    Can I find what the value is?

    thanks
    Well, if f is continuous and if the limit \lim_{n\to \infty}f^{n}(x) exists, it follows that such an x must be a fixed point of f: that is, x must satisfy the equation f(x)=x. In other words: to find those xs you solve the equation f(x)=x.
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