Let \(f(x) = \begin{cases x^3 + 8 & x<0 x^2 - 4 & x \ge 0 \end{cases}\) and \(g(x) = \begin{cases} (x-8)^{1/3} & x<0 (x+4)^{1/2} & x \ge 0 \end{cases}\) then find number of points of discontinuity of \(g(f(x))\).
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If \(g(x)\) is the inverse of \(f(x)\), then \(g(f(x)) = x\). A linear function \(y=x\) is continuous everywhere. Always check if functions are inverses before doing piecewise limit analysis.