Example 4 (1988 National High School League Question) There are three functions, the first one is $y=\varphi(x)$, its inverse function is the second function, and the graph of the third function is symmetrical to the graph of the second function about the line $x+y=0$, then the third function is ( ).
A. $y=-\varphi(x)$
B. $y=-\varphi(-x)$
C. $y=-\varphi^{-1}(x)$
D. $y=-\varphi^{-1}(-x)$
Solution: Choose B. Reason: The graph of the first function is symmetric to the graph of the second function about $x-y=0$, and the graph of the second function is symmetric to the graph of the third function about $x+y=0$, so the graph of the first function is symmetric to the graph of the third function about the origin.