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Math Help - Inverse functions

  1. #1
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    Inverse functions

    Show that f\left(g^{-1}(x)\right)=g\left(f^{-1}(x)\right) \Leftrightarrow f^{-1}\left(g(x)\right)=g^{-1}\left(f(x)\right)

    If we substitute x=f(u), we get

    f\left(g^{-1}\left(f(u)\right)\right)=g^{-1}\left(f\left(f^{-1}(u)\right)\right)
    \Updownarrow
    f\left(g^{-1}\left(f(u)\right)\right)=g(u)

    This means that g^{-1}\left(f(u)\right)=f^{-1}\left(g(u)\right), because f\left(f^{-1}\left(g(u)\right)\right)=g(u).

    Now, if the above equality holds for u, it must also hold for x, and we have that

    g^{-1}\left(f(x)\right)=f^{-1}\left(g(x)\right)

    and that

    f\left(g^{-1}(x)\right)=g\left(f^{-1}(x)\right) \Leftrightarrow f^{-1}\left(g(x)\right)=g^{-1}\left(f(x)\right)

    Which was to be shown.


    Does this hold, or have I overlooked something?

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by espen180 View Post
    Does this hold, or have I overlooked something?
    That is good. The only issue I have with this is that we do not know much about f,g themselves.
    If these were bijections on \mathbb{R} then what you did is okay.
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  3. #3
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    Yes, I assumed that was the case with f and g. But what if they were bijections on \mathbb{C}? Would it not hold then?
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