Closing fnf^n

Geometry Level 3

f(x)=x33xf(x)=x^3-3x

Find the closed form of fn(x),f^n(x), and then find the number of distinct real solutions of f16(x)=2.f^{16}(x)=2.

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


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

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