Question

The domain of $f\left(x\right)=\sqrt{x-2-2\sqrt{x-3}}-\sqrt{x-2+2\sqrt{x-3}}$, is

• A. $[3,5]$
• B. $(3,5)$
• C. $[5,\infty )$
• D. $[3,\infty )$

Answer 1

Answer: (D)

$[3,\infty )$

Given,

$f\left(x\right)=\sqrt{x-2-2\sqrt{x-3}}-\sqrt{x-2+2\sqrt{x-3}}$
Here $x-3\ge 0$
$\Rightarrow x\ge 3$
Also, $(x-2-2\sqrt{x-3})\ge 0$
$\Rightarrow x-2\ge 2\sqrt{x-3}$
$\Rightarrow (x-2{)}^2\ge 4(x-3)$
$\Rightarrow x^2-8x+16\ge 0$ is always true
$\Rightarrow x\ge 3$
Also, $(x-2+2\sqrt{x-3})\ge 0$
$\Rightarrow x-2\ge -2\sqrt{x-3}$
This is also always true as right hand side is always negative and left hand side is always positive for $x⩾3$
$\Rightarrow x\ge 3$
Hence, their union; $x\ge 3$ is the domain of $f\left(x\right)$