A rational function

Algebra Level 5

Suppose \(f(x)=\frac{p(x)}{q(x)},\) where \(p\) and \(q\) are polynomials with real coefficients, \(f\) is defined for all real numbers \(x\neq 1,\) and \(f(f(f(x)))=x,\) for all real \(x\) for which \(f(f(f(x)))\) is defined.

The smallest possible positive value of \(f(0)\) for such a function \(f\) can be written as \( \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?

Details and assumptions

You are not given if \(f(x) \) is defined at \( x = 1 \), or at any complex, non-real value.


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