Step 1 :\(f(g(x)) = (\sqrt{x+4})^2 - 1\)
Step 2 :\(f(g(x)) = x + 4 - 1 = x + 3\)
Step 3 :\(g(x) = \sqrt{x+4}\) is defined for all \(x\) such that \(x+4 \geq 0\), so \(x \geq -4\)
Step 4 :\(f(g(x)) = x + 3\) is defined for all real numbers. However, since \(g(x)\) is part of the composition, we must also consider the domain of \(g(x)\). Therefore, the domain of \(f \circ g\) is all \(x\) such that \(x \geq -4\)
Step 5 :\(\boxed{(f \circ g)(x) = x + 3}\)
Step 6 :\(\boxed{\text{Domain of } f \circ g: [-4, \infty)}\)