For $f(x)=\sqrt{x}$ and $g(x)=x+5$, find
a. $(f \circ g)(x)$;
b. the domain of $f \circ g$
a. $(f \oslash g)(x)=$

Step 1 ：The given functions are \(f(x) = \sqrt{x}\) and \(g(x) = x + 5\).

Step 2 ：The composition of two functions, denoted as \((f \circ g)(x)\), is the function obtained by applying \(f\) to the result of applying \(g\) to \(x\). That is, \((f \circ g)(x) = f(g(x))\).

Step 3 ：Substituting \(g(x)\) into \(f(x)\) gives us \((f \circ g)(x) = f(g(x)) = f(x + 5) = \sqrt{x + 5}\).

Step 4 ：The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For the function \(f(g(x)) = \sqrt{x + 5}\), the domain is all real numbers \(x\) such that \(x + 5 \geq 0\). Solving this inequality gives \(x \geq -5\). Therefore, the domain of \(f \circ g\) is \([-5, \infty)\).

Step 5 ：The division of two functions, denoted as \((f \oslash g)(x)\), is the function obtained by dividing \(f(x)\) by \(g(x)\).

Step 6 ：Dividing \(f(x)\) by \(g(x)\) gives us \((f \oslash g)(x) = f(x) / g(x) = \sqrt{x} / (x + 5)\).

Step 7 ：Final Answer: \(\boxed{(f \circ g)(x) = \sqrt{x + 5}}\), the domain of \(f \circ g\) is \(\boxed{[-5, \infty)}\), \(\boxed{(f \oslash g)(x) = \sqrt{x} / (x + 5)}\).