6. D. Reason: Let $y=f(x+1)$, its inverse function $f^{-1}(x)=x+1$, i.e., $x=f^{-1}(y)-1$. Therefore, the graph of the function $y=f(x+1)$ is symmetric to the graph of its inverse function $y=f^{-1}(x)-1$ about the line $y=x$.
6. (10th "Hope Cup" Invitational Competition Question) Given that $y=f(x)$ is a monotonic function defined on $\mathbf{R}$, then ( ).
A. The graph of the function $x=f^{-1}(y)$ is symmetric to the graph of $y=f(x)$ about the line $y=x$
B. The graph of the function $f(-x)$ is symmetric to the graph of $f(x)$ about the origin
C. The monotonicity of $f^{-1}(x)$ is opposite to that of $f(x)$
D. The graphs of the functions $f(x+1)$ and $f^{-1}(x)-1$ are symmetric about the line $y=x$