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If $f$ and $g$ are both linear, it's not difficult to find examples, e.g. $f(x)=7x-3$ and $g(x)=11x-5$; in general, for $f(x)=ax+b$ and $g(x)=cx+d$, the necessary and sufficient condition is easily seen to be $\frac{a-1}{c-1}=\frac{b}{d}$.

For non-linear examples, however, I've only been able to come up with relatively "trivial" examples, like $f(x)=\sqrt[3]{x}$ and $g(x)=x^{6}$, or $f(x)=-x$ and $g(x)=\frac{1}{x}$.

So my two-part question is:

(a) Are there any necessary or sufficient conditions so that for non-inverse, non-equal $f$ and $g$, $f\circ g = g\circ f$?

(b) Are there more complex examples than the above for which at least one of the two functions is non-linear? If so, I'd love any insight into how they were found.

A.J.
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    Presumably $f,g$ are real-valued functions on the reals? Are you imposing any other conditions. Eg take $f(x)=5$ for all $x$ and $g$ arbitrary except that $g(5)=5$. – almagest Sep 25 '19 at 10:06
  • Yes, I was asking the question about $\mathbb{R} \to \mathbb{R}$ functions, though I would certainly be open to $\mathbb{C} \to \mathbb{C}$ examples. Thanks for yours, it hadn't occurred to me; I would, though, have to include it in the same category as the other non-linear examples I mentioned. When I look at the linear example I gave, even though I know its derivation and that I can easily find other examples, it still seems far from obvious (to me) by inspection that it works; in a (admittedly vague) sense, that's what I meant by 'more complex' examples with non-linear functions. – A.J. Sep 26 '19 at 06:05

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