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What functions satisfy $f \circ g = g \circ f\ne \text{id}$ and $f \ne g$?

Some thoughts: This is fulfilled for $f(x) := \alpha x$ and $g(x) := \beta x$ for $\alpha, \beta \in \mathbb{R}$ such that $\alpha \beta \ne 1$ and $x \in \mathbb{R}$ because of commutativity of real numbers, but are there other not so trivial examples and how can they be characterised?

Edit to reflect how this shouldn't be a duplicate. The accepted answer on the suggested duplicate is very interesting and informative but only deals with polynomials / rational functions and there are a lot of other functions that could fit this description (and if not, why?).

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    $\begingroup$ You probably want $f \neq \text{id}$ and $g \neq \text{id}$ too. $\endgroup$
    – Chappers
    Commented Sep 8, 2019 at 14:39
  • $\begingroup$ Let $p$ be a point in the plane and $f$ and $g$ be any rotations around $p$. (Although this can be understood as a generalizion of your example, with $f(z)=p+(z-p)a$ and $g(z) = p+(z-p)b$.) $\endgroup$
    – MJD
    Commented Sep 8, 2019 at 14:44

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Your example is related to at least two others, $x+\alpha$ vs $x+\beta$ with $\alpha+\beta\ne0$ and $x^\alpha$ vs $x^\beta$ (where I'll leave you to work out suitable constraints on $\alpha,\,\beta$, bearing in mind what happens if $x<0$).

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A classic result of Ritt shows that polynomials that commute under composition must be, up to a linear homeomorphism, either both powers of $x$, both iterates of the same polynomial, or both Chebychev polynomials. Actually Ritt proved a more general rational function case - follow the link. His work was motivated by work of Julia and Fatou's work on Julia sets of rational functions, e.g. see here for a modern presentation.

A answer by Bill Dubuque on a similar question.

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