I attached the page from the Sequential III Amsco book that discusses restricted domains with regard to inverse functions. You'll notice that g^-1 notation is only used when referencing the inverse function and clearly omitted when referencing the inverse relation. However, in the same book, they do use g^-1 to denote an inverse relation but specifically say relation g^-1 every time they reference it.
There is another thing to keep in mind. The notation specifically used in #32 is not just f^-1 but is f^-1(x), which is clearly function notation. A function f has an input x and only one output, namely f(x). Using notation for the argument of the function implies that we are actually supposed to be working with a function.
Also, Amsco is not the final authority on all mathematics. I would personally put much more weight in the plethora of precalculus and calculus textbooks that define f^-1 to be the inverse function of f, as you mentioned. In terms of looking forward, a question like #32 would never be so ambiguous on a better-written test such as the AP Calc exams. The AP Calc exams denote f^-1 to be an inverse function; I think we ought to be consistent with the de facto standard for higher level math being taught in high schools.