The image of a recursive language under a computable function
Inspired by this question, I have come up with the following.

Assume that $\langle\cdot\rangle$ is an encoding for TM's, for which the encoding of one Turing machine is never a prefix of the encoding of another. 

Let $f$ be defined as the identity on an input which does not contain a TM enconding as a prefix. If the input contains such a prefix, $f$ returns this prefix.

$f$ is computable and it is well defined due to the encoding property.

Let $L=\left\{ \langle M\rangle n \mid \text{$M$ halts on the empty input after $n$ steps} \right\}$.

Then $L$ is recursive but $f(L)=\left\{ \langle M\rangle \mid \text{ $M$ halts on the blank tape} \right\}$ which is not recursive.