Because I can't find a function $h:\mathbb R\mapsto\mathbb R$ with the property $$h^{\circ 2}(x)=x^2+1$$ I'm looking for a function that almost has that property - that is, I would like to find a closed-form (and preferably elementary) function $h:\mathbb R\mapsto\mathbb R$ satisfying $$\lim_{x\t...
Let $s:\mathbb R\to\mathbb R$ be the function defined as $$s(x)=x+(-1)^{\lfloor x\rfloor}$$ Find a function $t:\mathbb R\to \mathbb R$ such that $t^{\circ 2}=s$, or prove that no such function exists. I was fairly sure that I could construct such a function $t$ by taking multiple cases a...
I've been doing a lot of research about functional half-iteration, and I posed the following question to myself: Consider the function $q:\mathbb R\mapsto\mathbb R$ defined as $$q(x)=x^2+1$$ Does $q^{\circ 1/2}$ exist? Does a continuous $q^{\circ 1/2}$ exist? What about a differentiable $...
Find the general $f^n(x)$ where $f^1(x)=e^x$ $f^{a}(f^{b}x)=f^{(a+b)}(x)$ where $n,x\in\mathbb R$ I'm fairly confident that no two $f^n(x)$ with different values of $n$ will intersect, since then we could use the argument that if $f^n(x)=x$ at some point $A$, then $f^m(x)=x$ a...
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