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Math Help - Finding inverse function for a mapping with a congruent class

  1. #1
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    Finding inverse function for a mapping with a congruent class

    Problem:

    F([X]7)=[2x]7

    Find the inverse for this function.

    So this function maps from x(mod 7) to 2x(mod 7). I was able to prove that the function is one-to-one and onto in a previous function, but I am lost after that.

    Am I looking for a function that produces [1]7 when multiplied with [2x]7?


    I really would appreciate any and all help.

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  2. #2
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    Re: Finding inverse function for a mapping with a congruent class

    No, there is no multiplication involved here. You are, as the problem says, looking for the inverse function.

    It's not at all difficult to write out exactly what f is:
    f(0)= 0, f(1)= 2, f(2)= 4, f(3)= 6, f(4)= 1, f(5)= 3, f(6)= 5.

    Therefore, f^{-1}(0)= 0, f^{-1}(1)= 4, f^{-1}(2)= 1, etc.
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: Finding inverse function for a mapping with a congruent class

    f(x)=2x\ (mod\ 7)

    f^{-1}(x)=2^{-1}x\ (mod\ 7)=4x\ (mod\ 7)
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    Re: Finding inverse function for a mapping with a congruent class

    Quote Originally Posted by alexmahone View Post
    f(x)=2x\ (mod\ 7)

    f^{-1}(x)=2^{-1}x\ (mod\ 7)=4x\ (mod\ 7)

    This is correct, but how did you go from 2^{-1}x\ (mod\ 7) to 4x\ (mod\ 7) ?
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  5. #5
    MHF Contributor alexmahone's Avatar
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    Re: Finding inverse function for a mapping with a congruent class

    Quote Originally Posted by hwill205 View Post
    This is correct, but how did you go from 2^{-1}x\ (mod\ 7) to 4x\ (mod\ 7) ?
    2\times 4=8\equiv 1 \pmod{7}

    So, 2^{-1}=4 \pmod{7}
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