Inverse Functions are certainly a minefield for students, and the situation is not helped by teachers’ use of sloppy language to describe the concept and the prescription of a mindless ritual for answering assigned questions.

This article points in the right direction although it’s not quite perfect in my opinion^{1}. But what got my friend Bruce to comment was one of the authors taking the objection to explanation by procedural prescription into another area where it might be less apt – namely the concept of average value.

Note 1:

Expressions like “the inverse of y=f(x)” are problematic because the relation defined by y=f(x) is the same as that defined by x=f^-1(y) and does have inverse relation defined by y=f^-1(x). So, contrary to the article, it is in some sense correct to say that the “the inverse of y=f(x) is y=f^-1(x)”, and the formal definition of functions as sets of ordered pairs does justify “switching x and y” if this is interpreted and explained properly.