What's the concept ??

Algebra Level 4

Let \(f(x)\) be an invertible function and \(g(x)\) be its inverse function . It is given that \(f(x)=x\) for only one value of \(x\) in the range of \(g(x).\)

What is the number of values of \(x\) in the domain of \(f(x)\) for which \(f(x)-g(x)=n-\lfloor \frac{2015}{4} \rfloor\) ? ,where \(n\) is the only natural number for which the polynomial \(1+x+x^2+x^3+...+x^{n-1}\) divides the polynomial \(1+x^2+x^4+...+x^{2010}.\)

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