Find $(f \circ g)(x)$ and $(g \circ f)(x)$ for the pair of functions. \[ f(x)=\frac{x+3}{2}, g(x)=\sqrt{1-x} \]

Step 1 ：Given the functions \(f(x)=\frac{x+3}{2}\) and \(g(x)=\sqrt{1-x}\)

Step 2 ：We are asked to find the compositions \(f \circ g(x)\) and \(g \circ f(x)\)

Step 3 ：For \(f \circ g(x)\), we first apply the function \(g\) to \(x\), and then apply the function \(f\) to the result. This gives us \(f(g(x)) = f(\sqrt{1 - x}) = \frac{\sqrt{1 - x}+3}{2}\)

Step 4 ：For \(g \circ f(x)\), we first apply the function \(f\) to \(x\), and then apply the function \(g\) to the result. This gives us \(g(f(x)) = g(\frac{x+3}{2}) = \sqrt{1-\frac{x+3}{2}}\)

Step 5 ：\(\boxed{(f \circ g)(x) = \frac{\sqrt{1 - x}}{2} + \frac{3}{2}}\)

Step 6 ：\(\boxed{(g \circ f)(x) = \sqrt{-\frac{x}{2} - \frac{1}{2}}}\)