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Examine the existence of a function $f:\mathbb R\to \mathbb R$ such that:

$\forall x \in \mathbb R\ :f(f(x))= x^2 +1$

How to approach this type of problems? I study functional equation from a book of B.J.Venkatachala but I don't know how to approach problems having a composition of functions. Need help.

Oussama Boussif
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Dots_and_Arrows
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    https://www.quora.com/If-f-f-x-x-2-1-what-is-f-x –  May 17 '17 at 16:52
  • A small bait-and-switch: Like all logistic maps, this one is solved numerically through Schröder's equation. However, there is a concise closed solution for $f(f(x))=x^2-2$, instead, invented by him in 1870: here. – Cosmas Zachos Jul 13 '17 at 22:31

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This is an interesting problem. This paper discusses the problem at length and contains a strong result that implies, in particular, the following:

Let $g:\mathbb{R}\longrightarrow\mathbb{R}$ be defined by $g(x)=ax^2+(b+1)x+c$ and let $\Delta(g)=b^2-4ac$. Then for each positive integer $n$, a solution $f:\mathbb{R}\longrightarrow\mathbb{R}$ to $f^n=g$ exists if and only if $\Delta(g)\leq 1$.

In your case, we have $g(x)=x^2+1$, so $\Delta(g)=1-4=-3$, and hence existence is guaranteed.

Further interesting readings on the topic might include Schröder's equation, or this paper on solutions of $f^2=g$ for $g$ that fail to be injective.

Fimpellizzeri
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