Answer
$f+g,f-g,fg$: for all Domain=$\{ x|x=2\}$.
$\frac{f}{g}$: the domain is the empty set.
Work Step by Step
$f(x)=\sqrt {x-2}$ and $g(x)=\sqrt {2-x}$.
$f(x)+g(x)=\sqrt {x-2} + \sqrt {2-x}$. To find the domain of the sum function, $x-2\geq0$, $x\geq2$ and $2-x\geq0$, $x\leq2$. Therefore,
Domain=$\{ x:x=2\}$.
$f(x)-g(x)=\sqrt {x-2} - \sqrt {2-x}$. To find the domain of the difference function, $x-2\geq0$, $x\geq2$ and $2-x\geq0$, $x\leq2$. Therefore,
Domain=$\{ x:x=2\}$.
$f(x) \times g(x)=\sqrt {x-2} \times \sqrt {2-x}=\sqrt {(x-2)(2-x)}$. To find the domain of the product function, $x-2\geq0$, $x\geq2$ and $2-x\geq0$, $x\leq2$. Therefore,
Domain=$\{ x:x=2\}$.
$\frac{f(x)}{g(x)}=\frac{\sqrt {x-2}}{\sqrt {2-x}}=\sqrt {\frac{x-2}{2-x}}$.To find the domain of the quotient function, $x-2\geq0$, $x\geq2$ and $2-x\gt0$, $x\lt2$ .Therefore,
Domain=$\{ x:x=\{\}\}$. The domain is the empty set.