Find the domain of the function \[f(x)=\sqrt{1-\sqrt{2-\sqrt{3 - \sqrt{x}}}}.\]

Answer 1

Since all the square roots in the expression must be real, we must have \[3 - \sqrt{x} \ge 0,\]so $\sqrt{x} \le 3,$ or $x \le 9.$ Also, we must the expression inside the square root to be nonnegative, so \[2 - \sqrt{3 - \sqrt{x}} \ge 0,\]which turns into $\sqrt{3 - \sqrt{x}} \le 2.$ Squaring, we get $3 - \sqrt{x} \le 4,$ so $\sqrt{x} \ge -1.$ Since we must also have $\sqrt{x} \le 3,$ we must have $\sqrt{x} \le 3$ and $\sqrt{x} \ge 0.$ Hence, $0 \le x \le 9.$ The domain of $f(x)$ is $\boxed{[0,9]}.$