Not another composed function

Algebra Level 4

Define the function \(f(x)=\frac{2x}{1-x^2}\). Find the number of distinct real solutions of the equation \(f^{(5)} (x) =x.\)

Details and assumptions

\( f^{(n)} (x) \) denotes the function \(f\) applied \(n\) times. In particular, \( f^{(5)} (x) = f(f(f(f(f(x)))))\).

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