  It is well known that the entropy $H(X)$ of a discrete random variable $X$ is
always greater than or equal to the entropy $H(f(X))$ of a function $f$ of $X$,
with equality if and only if $f$ is one-to-one. In this paper, we give tight
bounds on $H(f(X))$ when the function $f$ is not one-to-one, and we illustrate
a few scenarios where this matters. As an intermediate step towards our main
result, we derive a lower bound on the entropy of a probability distribution,
when only a bound on the ratio between the maximal and minimal probabilities is
known. The lower bound improves on previous results in the literature, and it
could find applications outside the present scenario.
