7.82 The graph of the function that is symmetric to the graph of $y=f(x-a)+b$ with respect to the line $y=x$ is
(A) $y=f^{-1}(x-a)+b$.
(B) $y=f^{-1}(x+a)-b$.
(C) $y=f^{-1}(x-b)+a$.
(D) $y=f^{-1}(x-b)-a$.
(China Beijing High School Mathematics Competition, 1992)
[Solution] The function corresponding to the image symmetrical to the graph of $y=f(x-a)+b$ with respect to the line $y=x$ should be
$$
x=f(y-a)+b \text {, i.e., } x-b=f(y-a) \text {, }
$$

then $y-a=f^{-1}(x-b)$, or $y=f^{-1}(x-b)+a$ is the desired function. Therefore, the answer is $(C)$.