6. Given the function $y=-\frac{x-a}{x-a-1}$, the graph of its inverse function is symmetric with respect to the point $(-1,3)$. Then the real number $a$ equals ( ).
(A) 2
(B) 3
(C) -2
(D) -3
6 Method one: Take a point $\left(1, \frac{1}{a}-1\right)$ on the original function graph, then the point $\left(\frac{1}{a}-1,1\right)$ is on the graph of its inverse function. The point symmetric to it about the point $(-1,3)$ is $\left(-1-\frac{1}{a}, 5\right)$, thus the point $\left(5,-1-\frac{1}{a}\right)$ is on the original function graph, so we have
$$
-1-\frac{1}{a}=-\frac{5-a}{5-a-1},
$$

Solving for $a$ gives $a=2$. Therefore, the answer is A.
Method two: Since the function $y=-\frac{x-a}{x-a-1}$ and its inverse function's graphs are symmetric about the line $y=x$, their centers of symmetry must also be symmetric about $y=x$. The point $(-1,3)$ is symmetric to the point $(3,-1)$ about the line $y=x$. Since the function $y=-\frac{x-a}{x-a-1}$, or
$$
y=-1+\frac{1}{x-(a+1)},
$$

is symmetric about the point $(a+1,-1)$, we have $a+1=3$, giving $a=2$. Therefore, the answer is A.