Question

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 \( g(f(x)) \).

Hint:

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.

Answer 1