3. For any function $y=f(x)$, in the same Cartesian coordinate system, the graphs of the functions $y=f(x-1)$ and $y=f(-x+1)$ are always
A. symmetric with respect to the $x$-axis
B. symmetric with respect to the line $x=1$
C. symmetric with respect to the line $x=-1$
D. symmetric with respect to the $y$-axis
Solve:
Let $x-1=-x+1 \Rightarrow x=1$, thus the graph of the function $y=f(x-1)$ is symmetrical to the graph of the function $y=f(-x+1)$ about the line $x=1$. Therefore, the answer is $B$.
Note: The graph of the function $y=f(x)$ is symmetrical to the graph of the function $y=f(-x)$ about the $y$-axis, the graph of the function $y=f(x-1)$ is obtained by shifting the graph of the function $y=f(x)$ to the right by 1 unit, and the graph of the function $y=f(-x+1)=$ $f[-(x-1)]$ is obtained by shifting the graph of the function $y=f(-x)$ to the right by 1 unit. Therefore, the graph of the function $y=f(x-1)$ is symmetrical to the graph of the function $y=f(-x+1)$ about the line $x=1$.