Inverse Function: unique?

Question

Is it true in general that the inverse of a function is unique if it exists? Why is this so?

Clearly inverses in groups are unique. However, that seems not directly applicable in this case...

Answer 1

If it is a two-sided inverse, then yes. To see why assume that $f_1,f_2$ are both two-sided inverses of $h$. Then $$f_1 = f_1\circ (h\circ f_2) = (f_1\circ h)\circ f_2 = f_2$$

Note that this proof implies that if $h$ has a left inverse and a right inverse, then they are necessarily the same.

Answer 2

$x^2=1\iff x=\pm1;\quad\sin x=0\iff x=2k\pi,\quad k\in\mathbb Z.~$ So no, they are not necessarily unique, especially if the function is periodic or even, etc. If you do want uniqueness, however, then you have to separate the domain of definition into intervals of bijectivity.