Closing $f^n$

Geometry Level 3

$f(x)=x^3-3x$

Find the closed form of $f^n(x),$ and then find the number of distinct real solutions of $f^{16}(x)=2.$

Note: $f^n(x)$ is the $n^\text{th}$ composition of $f,$ that is, $f^n(x) = \underbrace{f \circ f \circ \cdots \circ f}_{n \text{ times}}(x).$

Bonus: If $g(x)=\sqrt{2+x},$ find the closed form of $g^n(x).$

Closing fnf^nfn