Answer
$f+g,f-g,fg$: for all Domain=$\{ x|x=5\}$.
$\frac{f}{g}$: Domain=$\{ x|x=\{\}\}$
Work Step by Step
$f(x)=\sqrt {x-5}$ and $g(x)=\sqrt {5-x}$.
$f(x)+g(x)=\sqrt {x-5} + \sqrt {5-x}$. To find the domain of the sum function, the domain of $f(x)$ is $x-5\geq0$, $x\geq5$ and the domain of $g(x)$ is $5-x\geq0$, $x\leq5$. Therefore,the domain of $f+g$ is the intersection of domain of $f$ and domain of $g$.
Domain=$\{ x:x=5\}$.
$f(x)-g(x)=\sqrt {x-5} - \sqrt {5-x}$. To find the domain of the difference function, the domain of $f(x)$ is $x-5\geq0$, $x\geq5$ and the domain fo $g(x)$ is $5-x\geq0$, $x\leq5$. Therefore, the domain of $f-g$ is the intersection of domain of $f$ and domain of $g$.
Domain=$\{ x:x=5\}$.
$f(x) \times g(x)=\sqrt {x-5} \times \sqrt {5-x}=\sqrt {(x-5)(5-x)}$. To find the domain of the product function, the domain of $f(x)$ is $x-5\geq0$, $x\geq5$ and the domain of $g(x)$ is $5-x\geq0$, $x\leq5$. Therefore, the domain of $fg$ is the instersection of the domain of $f$ and domain of $g$.
Domain=$\{ x:x=5\}$.
$\frac{f(x)}{g(x)}=\frac{\sqrt {x-5}}{\sqrt {5-x}}=\sqrt {\frac{x-5}{5-x}}$.To find the domain of the quotient function,the domain of $f$ is $x-5\geq0$, $x\geq5$ and the domain of $g$ $5-x\gt0$, $x\lt5$ .Therefore, the domain of $\frac{f}{g}$ is the intersection of the domain of f and the domain of $g$
Domain=$\{ x:x=\{\}\}$. The domain is empty set.
