Given functions $f$ and $g$, find $(a)(f \circ g)(x)$ and its domain, and $(b)(g \circ f)(x)$ and its domain. \[ f(x)=\sqrt{x}, g(x)=x+11 \]

Step 1 ：Given functions $f$ and $g$, where $f(x)=\sqrt{x}$ and $g(x)=x+11$.

Step 2 ：To find the composition of functions and their domains, we need to substitute one function into the other.

Step 3 ：For $(f \circ g)(x)$, we substitute $g(x)$ into $f(x)$, which gives us $\sqrt{x+11}$. The domain of $(f \circ g)(x)$ is the set of all real numbers $x$ for which $(f \circ g)(x)$ is defined, i.e., $x \geq -11$.

Step 4 ：For $(g \circ f)(x)$, we substitute $f(x)$ into $g(x)$, which gives us $\sqrt{x} + 11$. The domain of $(g \circ f)(x)$ is the set of all real numbers $x$ for which $(g \circ f)(x)$ is defined, i.e., $x \geq 0$.

Step 5 ：Final Answer: \(\boxed{(f \circ g)(x) = \sqrt{x+11}, \text{domain} = [-11, \infty)}\) and \(\boxed{(g \circ f)(x) = \sqrt{x} + 11, \text{domain} = [0, \infty)}\)