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

(a) [4, ∞)
(b) (−∞, 4]
(c) (4, ∞)
(d) (−∞, 4)

Answer 1

(a) [4, ∞)

$f\left(x\right)=\sqrt{x-3-2\sqrt{x-4}}-\sqrt{x-3+2\sqrt{x-4}}$

$$\begin{gathered} \mathrm{For}f\left(x\right)\mathrm{to}\mathrm{be}\mathrm{defined},x-4\ge 0 \\ \Rightarrow x-4\ge 0 \\ \Rightarrow x\ge 4....\left(1\right) \\ \mathrm{Also},x-3-2\sqrt{x-4}\ge 0 \\ \Rightarrow x-3-2\sqrt{x-4}\ge 0 \\ \Rightarrow x-3\ge 2\sqrt{x-4} \\ \Rightarrow (x-3{)}^2\ge {\left(2\sqrt{x-4}\right)}^2 \\ \Rightarrow x^2+9-6x\ge 4\left(x-4\right) \\ \Rightarrow x^2-10x+25\ge 0 \\ \Rightarrow (x-5){}^2\ge 0,\mathrm{which}\mathrm{is}\mathrm{always}\mathrm{true}. \\ \mathrm{Similarly},x-3+2\sqrt{x-4}\ge 0\mathrm{is}\mathrm{always}\mathrm{true}. \\ \mathrm{Thus},\mathrm{dom}(\mathrm{f}\left(\mathrm{x}\right))=[4,\infty ) \end{gathered}$$