Question

The domain of $f\left(x\right)=\sqrt{x-4-2\sqrt{x-5}}-\sqrt{x-4+2\sqrt{x-5}}$ is

• A. $(5,\infty )$
• B. $[5,\infty )$
• C. $(4,\infty )$
• D. $[4,\infty )$

Answer 1

The correct option is B $[5,\infty )$

$f\left(x\right)=\sqrt{x-4-2\sqrt{x-5}}-\sqrt{x-4+2\sqrt{x-5}}$
$=\sqrt{x-5-2\left(\sqrt{x-5}\right)+1}-\sqrt{x-5+2\left(\sqrt{x-5}\right)+1}=\sqrt{{(\sqrt{x-5}-1)}^2}-\sqrt{{(\sqrt{x-5}+1)}^2}=|\sqrt{x-5}-1|-|\sqrt{x-5}+1|$

Now, for $f\left(x\right)$ to be defined,
$x-5\ge 0\Rightarrow x\in [5,\infty )$