You needn't be confused. Things are just wrong, sometimes. That "correct answer" is not correct at all - not all that related, frankly. Answer to a different problem, maybe? The inverse FUNCTION may not exist, but there is no need to throw out everything that may be called an "inverse".
You can prove the existence of the inverse mechanically.
1) Draw your function, y = f(x) on some thin paper.
2) On the same set of coordinate axes, add the line y = x and extend it until it hits the borders of the paper in both directions.
3) With the graph facing you, grab the line y = x at both points where it leaves the edge of the paper.
4) Flip the paper half way over, with the lower right corner ending up in the upper left corner and the graph now facing away from you..
5) Hold the paper up to the light and there it is..
No need to claim that it doesn't exist.