Example 1 Given $f(x)=\sqrt{25-x^{2}}(0 \leqslant x \leqslant 4)$, find the inverse function of $f(x)$.
Since $0 \leqslant x \leqslant 4$, we have $0 \leqslant x^{2} \leqslant 16, 9 \leqslant 25-x^{2} \leqslant 25$, thus $3 \leqslant y \leqslant 5$. Also, because $y=\sqrt{25-x^{2}}, y^{2}=25-x^{2}$, we have $x^{2}=25-y^{2}$. Since $0 \leqslant x \leqslant 4$, we get $x=$ $\sqrt{25-y^{2}}(3 \leqslant y \leqslant 5)$. By swapping $x$ and $y$, we have $y=f^{-1}(x)=\sqrt{25-x^{2}}(3 \leqslant x \leqslant 5)$.