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It is well known that the entropy $H(X)$ of a finite random variable isalways greater or equal to the entropy $H(f(X))$ of a function $f$ of $X$, withequality if and only if $f$ is one-to-one. In this paper, we give tights boundson $H(f(X))$ when the function $f$ is not one-to-one, and we illustrate a fewscenarios where this matters. As an intermediate step towards our main result,we prove a lower bound on the entropy of a probability distribution, when onlya bound on the ratio between the maximum and the minimum probability is known.Our lower bound improves previous results in the literature, and it could findapplications outside the present scenario.

$H(X)$ vs. $H(f(X))$