2. Given the function $y=f(x)$ has an inverse function $y=$ $f^{-1}(x)$. If the graph of $y=f(x)$ is rotated $90^{\circ}$ counterclockwise around $(1,-1)$, the equation of the resulting curve is ( ).
(A) $y=f^{-1}(-x)-2$
(B) $y=-f^{-1}(-x)-2$
(C) $y=f^{-1}(-x+1)-1$
(D) $y=f^{-1}(-x-1)+1$
2. A.

The point $(t, f(t))$ is rotated $90^{\circ}$ counterclockwise around $(1, -1)$, resulting in $(-f(t), t-2)$.
Let $x = -f(t)$. Then
$$
y = t-2 = f^{-1}(-x) - 2 \text{.}
$$