# Suitable function H(H(x))=H(x)

• Apr 10th 2011, 04:03 PM
Conn
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! :)
• Apr 10th 2011, 06:38 PM
theodds
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.