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0
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Compute the domain of the real-valued function $$f(x)=\sqrt{3-\sqrt{5-\sqrt{x}}}$$

Aug 11, 2022

#1
+118211
+1

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.

Aug 11, 2022
#2
+33338
+1

Here's a graphical check showing the correctness of Melody's answer.

Aug 11, 2022
#3
+118211
0

Thanks Alan

Melody  Aug 12, 2022