Do inverse operation undo each other?

Answer 1

Usually, but not always: it depends on the domains and codomains.

Any function that is many-to-one (for example all even powers, all trigonometric functions) have an inverse operation that is defined over a restricted domain. They will, therefore, return the principal value but not necessarily the original value.

A couple of simple example, using the square and square root functions:

(-2) squared = 4

but

sqrt(4) = +2, not -2.

sin(150°) = 0.5

but

sin-1(0.5) = 30°

It is, of course, possible to define the sqrt function so that it returns the negative root, but then it will not return the positive one.