# 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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