Question

Why does it not make sense to graph f and f−1 on the same set of axes?

Answer 1

Here you have two graphs, one -- graph of the function y=f(x) on the coordinate plane with axes x and y, another -- graph of the function $x=f^{-1}(y)$ on the coordinate plane with axes y and x.

Actually, these graphs are graphs of the same function, only axes are changed.

It makes the sense to graph to functions  $f(x)$ and $f^{-1}(x)$ on the same set of axes, not graphs of the same function $y=f(x)$ and $x=f^{-1}(y)$ on the different set of axes.