Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

18, Find the domain of the independent variable $x$ for the following functions:
(1) $y=\frac{1}{\sqrt{|x|-2}}$ ;
(2) $y=\frac{1}{\lg \sqrt{x-2}}$ ;
(3) $y=\frac{1}{3-\sqrt{x-2}}$ ;
(4) $y=\frac{1}{| \lg | x-2 |}$ ;
(5) $y=\frac{\lg (x-1)}{2-\sqrt{x}}$ ;
(6) $y=\sqrt{2-x} \div \frac{1}{\sqrt{x+1}}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

(Ans: (1) $x>2$ and $x2$ and $x \neq 3$ , (3) $x \geqslant 2$ and $x \neq 11$ , (4) $x \neq 1,2,3$ , (5) $x>1$ and $x \neq 4$ , (6) $-1<x \leqslant 2$ )

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