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 \(\boxed{[-5, \infty)}\), \(\boxed{(f \oslash g)(x) = \sqrt{x} / (x + 5)}\).