$7 \cdot 86$ The function $y=f(x)$ is defined on $R$, and it has an inverse function. If the graph of the function is rotated $90^{\circ}$ counterclockwise around the origin in the $x O y$ plane, the new graph represents the function
(A) $y=-f^{-1}(-x)$.
(B) $y=-f^{-1}(x)$.
(C) $y=f^{-1}(x)$.
(D) $y=f^{-1}(-x)$.
(2nd "Hope Cup" National Mathematics Invitational Competition, 1991)
[Solution] Note that the graph of the function $y=f(x)$ is symmetric to the graph of $x=f(y)$ with respect to the line $y=x$.
The graph of the function $-x=f(y)$ is symmetric to the graph of $x=f(y)$ with respect to the $y$-axis.
Therefore, the graph of the function $-x=f(y)$ can be regarded as the result of rotating the graph of the function $y=f(x)$ counterclockwise by $90^{\circ}$ around the origin $O$ in the $x O y$ plane.
Also, $-x=f(y)$ is usually written as $\quad y=f^{-1}(-x)$.
Hence, the answer is $(D)$.