2023/24 AHSMC Part 1 Problem 9

Write $f(x) = (x+4)^2 - 3$, so $f(x) \ge -3$ for any real $x$. The function $f$ is increasing on $[-4,\infty)$, hence $f(f(x)) \ge f(-3) = -2$, $f(f(f(x))) \ge f(-2) = 1$, $f(f(f(f(x)))) \ge f(1) = 22$, and $f(f(f(f(f(x))))) \ge f(22) = 673$. This lower bound is attained at $x = -4$. The answer is (E).