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Comment on uniqueness

A function has another function as inverse if each of its values occur only once.

Otherwise, if say f(a) = y and f(b) = y, we have a problem deciding if f - 1(y) = a or f - 1(y) = b.

Thus x2 = y2 is satisfied by x = y and x = -y.

We can resolve this difficulty if it occurs by defining the inverse to be some specific value that obeys our conditions.

This is what we do for the square root. is the positive square root but - is also a square root of x. Thus the "square" has a two-valued inverse.

In the example above, we would have f-1(y) =  a and also b.

If you want to define a single-valued function f - 1, you must add another condition. For the function , that condition is .