Say we have a one-to-one (total) function $f:\mathbb{N}\to\mathbb{N}$ and a Turing-machine $T_f$ that computes it. Suppose further that $T_f$ runs in polynomial time wrt. length of the input.

Are there functions $f$ that are computable in polynomial time but whose inverse is known not to be computable in polynomial time?

Does the situation change if we drop the one-to-one requirement and define the inverse as, say, min$(f^{-1})$? How about if we change the complexity class in question?

honesty- that the length of the output must be nearly equal to some polynomial of the length of the input. As Joel's example shows, this requirement is essential... $\endgroup$ – François G. Dorais♦ Jun 21 '11 at 11:39