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Question
The domain of definition of \[f\left( x \right) = \sqrt{x - 3 - 2\sqrt{x - 4}} - \sqrt{x - 3 + 2\sqrt{x - 4}}\] is
Options
(a) [4, ∞)
(b) (−∞, 4]
(c) (4, ∞)
(d) (−∞, 4)
Solution
(a) [4, ∞) \[f\left( x \right) = \sqrt{x - 3 - 2\sqrt{x - 4}} - \sqrt{x - 3 + 2\sqrt{x - 4}}\]
\[\text{ For f(x) to be defined } , x - 4 \geq 0\]
\[ \Rightarrow x - 4 \geq 0\]
\[ \Rightarrow x \geq 4 . . . . (1)\]
\[\text{ Also} , x - 3 - 2\sqrt{x - 4} \geq 0\]
\[ \Rightarrow x - 3 - 2\sqrt{x - 4} \geq 0\]
\[ \Rightarrow x - 3 \geq 2\sqrt{x - 4}\]
\[ \Rightarrow (x - 3 )^2 \geq \left( 2\sqrt{x - 4} \right)^2 \]
\[ \Rightarrow x^2 + 9 - 6x \geq 4\left( x - 4 \right)\]
\[ \Rightarrow x^2 - 10x + 25 \geq 0\]
\[ \Rightarrow (x - 5) {}^2 \geq 0, \text{ which is always true .} \]
\[\text{ Similarly,} x - 3 + 2\sqrt{x - 4} \geq 0 \text{ is always true } . \]
\[\text{ Thus, dom } (f(x)) = [4, \infty )\]
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