Example 2. Find the inverse function of $y=3+\sqrt{x-1} \quad(x \geq 1)$.
Solve for the range: $\because x=1, \therefore y-3=\sqrt{x}-1$ $\geqslant 0$, i.e., $y \geqslant 3$.
Inverse solution: From $y-3=\sqrt{x-1}$, we get $x=(y-3)^{2}+1$.
Interchange: $y=(x-3)^{2}+1, x \geqslant 3$ is the required inverse function.