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.
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inverse operation
Operation, and inverse operation
Inverse operations. Additive inverse is not one operation but they are elements of a set.
inverse operations
Inverse operations.