Compute the domain of the real-valued function \(f(x)=\sqrt{3-\sqrt{5-\sqrt{x}}}\)
everything under the square roots must be greater or equal to 0
so,
\(1)\quad x\ge0\\ 2)\quad 5-\sqrt x \ge0\\ \qquad 5\ge \sqrt x\\ \qquad \sqrt x \le 5\\ \qquad 0\le x \le 25\\ 3)\quad \sqrt{5-\sqrt x }\le 3\\ \qquad 0 \le 5-\sqrt x \le 9\\ \qquad -5 \le -\sqrt x \le 4\\ \qquad -5 \le -\sqrt x\qquad and \qquad -\sqrt x \le 4\\ \qquad \sqrt x \le 5\qquad \qquad and \qquad \sqrt x \ge -4\\ \qquad x \le 25\qquad \qquad and \qquad x \ge -0\\ so\\ domain \;\;[0,25] \)
I have not checked my answer, you need to do that.
I would also like a response from you, last time you did not give me one.