Why does a function and its inverse always intersect on the line y=x I've been working through a textbook and noticed that if a function intersects with its inverse function it's always on the line y = x. Why is this?
Well, the statement is not true. Take any bijective function for which $f(1) = 2$ and $f(2)=1$ - and just make the graph not be symmetric in the diagonal ($y=x$). Then the graph of $f$ and that of $f^{-1}$ may intersect on the diagonal, but they will certainly also meet at $(1,2)$ and at $(2, 1)$. (Sorry for the poor drawing...) By the way, the condition "not symmetric in the diagonal" is not really necessary; at the extreme, the function $f(x)=-x$ is its own inverse, so it intersects itself at all the points on it. To avoid such brain twisters, just assume the graphs only intersect in a few points.
If $f(5) = 22$ then $f^{-1}(22) = 5$. $\qquad\longleftarrow$This is the main idea, without which this will be incomprehensible.

Similarly:

If $f(9) = 9$ then $f^{-1}(9) = 9$. And the point $(9,9)$ is on the line  $y=x$, since $9=9$.