How to get $f(x)$, if we know that $f(f(x))=x^3+x$? Is there an elementary function $f(x)$ that satisfies the equation?
Reference: How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?
Note: Will Jagy gave a superb approximation of $f(f(x))$ in this reference above, that is this even for large x>10.
Now, how do we derive $f(x)$ in $f(f(x))$ which give good approximation using the same method for x>10?
It's messier for large $x$, but still the first couple of terms are gettable.
The first term is $x^{\sqrt3}$ of course.
Next try $x^{\sqrt3}+\alpha(x)+\ldots$. Then
$$f(f(x))=f\left(x^{\sqrt3}(1+\frac{\alpha(x)}{x^{\sqrt3}}+\ldots)\right)\\
=x^3(1+\sqrt3 \frac{\alpha(x)}{x^{\sqrt3}}+\ldots) +\alpha(x^{\sqrt3})+\ldots$$
Pick an $\alpha(x)$ so the leading correction term is $1x^1$. You can then introduce a $\beta(x)$ that will remove the next-largest correction term, and so on.
