# The Permutational Composition

Algebra Level 5

Given $f(x)=\frac{\sqrt{3x^2+1}}{x}.$ We define a positive algebraic function $$g$$ such that the composition $$fog$$ satisfies the equation $\prod_{r=1}^n (fog(r))^2=^{n+3}P_n$ for every positive integer $$n$$. Find $$4g(4)$$.

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