Polynomials of polynomials

Level pending

Let \(f\) be a (possibly constant) polynomial such that for all real \(x\):

\(f(f(x) + f(-x)) = 9f(x^2)\)

The largest possible value of \(f(9)\) can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. Find \(ab\).

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