Question

The domain of the function $f\left(x\right)=\sqrt{x-3-2\sqrt{x-4}}-\sqrt{x-3+2\sqrt{x-4}}$ is

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

Answer 1

The correct option is A $[4,\infty )$

Given,

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

for $f\left(x\right)$ to be defined we need to have

$x-3-2\sqrt{x-4}\ge 0-\left(1\right)x-3+2\sqrt{x-4}\ge 0-\left(2\right)x-4\ge 0-\left(3\right)$

Solving $\left(1\right)$

$x-3\ge 2\sqrt{x-4}\Rightarrow x^2+9-6x\ge 4(x-4)\Rightarrow x^2+9-6x\ge 4x-16\Rightarrow x^2-10x+25\ge 0\Rightarrow {(x-5)}^2\ge 0$

Solving $\left(2\right)$

$x-3\ge -2\sqrt{x-4}\Rightarrow x^2+9-6x\ge 4\sqrt{x-4}\Rightarrow {(x-5)}^2\ge 0$

Solving $\left(3\right)$

$x-4\ge 0\Rightarrow x\ge 4$

So the solution will be the intersection of the three regions found above which is

$xϵ[4,\infty )$