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. $[4,\infty )$
• B. $(-\infty ,4]$
• C. $(4,\infty )$
• D. $(\infty ,4)$

Answer 1

The correct option is A

$[4,\infty )$

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