I have to construct a non-monotonic function (defined on any interval) such that $f\circ f(x)=9x+4$. I tried making a $2$ branch function with $f(x)=3x+1$ and $f(x)=-3x-2$, but I don't know how to choose the intervals. Can someone please give me an idea?
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Splicing your two monotonic solutions you have will not result in compatible domain functions, as they are analogous to the $\pm y$ of Babbage's equation. You may shift the distracting fixed point x =-1/2 to 0 by $g(z+1/2)\equiv f(z)+1/2$, so that $g(g(y))=9y$, for $y\equiv x+1/2$. – Cosmas Zachos Oct 06 '18 at 15:40