I think when you say "the operation", you mean "the function"; and you are saying that the same thing you do to get the output, you can do to the output to get the input back. That's right.

When you said "f(x) = ff(x)", you meant either "ff(x) = x" or "f^-1(x) = f(x)". Right?

Yes, I am pointing out incorrect language because precision is important in math, and the way to learn it, unfortunately, is to be caught saying something you don't mean, and gradually learn to catch it yourself so other don't have to. It's more or less the same idea as proofreading what you write before hitting Send, rather than after.

Yes, the graph of y = 1/x is **discontinuous **(it consists of two "branches"), which is not specifically relevant here, and it is **symmetrical **with respect to the line y=x, which is exactly the point. Your other example, y = a - x, is in fact **perpendicular **to that line; can you see how that makes it symmetrical, and therefore an involution?

A line that is **parallel **to y=x will not be symmetrical, unless it is the line y=x. The Wikipedia article I referenced in my first response gives some other examples. Another, more subtle, is y = (2x + 1)/(3x - 2).