Steve's Mapping
The two functions are
$$y=\frac{2x+9}{x+2}\quad\quad x=\frac{9-2y}{y-2}$$
Note that these have been chosen to be inverse function and also to
have whole number fixed points and dual values on the given
ranges
So, points of interest are $(3, 3)$ and $(-7, 1)$.
The symmetry of the graphs should emerge from plotting, although we
have chosen to restrict to integer values for the initial detective
work part of the problem
Good luck and happy hunting