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Thread: Suitable function H(H(x))=H(x)

  1. #1
    Junior Member
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    Oct 2010
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    Suitable function H(H(x))=H(x)

    Hi just a question from the book I'm a bit puzzled with, probably fairly basic but I am very tired and can't make any sense of it :/

    Find a suitable function $\displaystyle H$ such that $\displaystyle H(H(x))=H(x)$, and $\displaystyle H(1)=36, H(2)=\frac{\pi}{3}, H(13)=47, H(36)=36, H(\frac{\pi}{3})=\frac{\pi}{3}, H(47)=47$

    Hint: Don't try to solve for $\displaystyle H(H(x))=H(x)$, the further conditions are to guide you to a suitable $\displaystyle H$

    I understand how they go together as in:

    $\displaystyle H(H(1))=H(1)$
    $\displaystyle H(H(2))=H(2)$
    $\displaystyle H(H(13)=H(13)$

    But I'm not sure of the form the function $\displaystyle H$ will take??

    Thank you!
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  2. #2
    Senior Member
    Joined
    Oct 2009
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    The important values of $\displaystyle x$ are the ones where $\displaystyle H(x) = x$. For such an $\displaystyle x$, if $\displaystyle y$ is such that $\displaystyle H(y) = x$ then clearly $\displaystyle H(H(y)) = H(y)$. Think along the lines of partitioning the reals.
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