jmc

Note that if $0\le x<1$, then $\lfloor x\rfloor =0$, so $f(x)=-x$. Therefore, the range of $f(x)$ includes the interval $(-1,0]$. This is in fact the whole domain; $f(x)$ can't be less than or equal to $-1$, because $x$ and $\lfloor x\rfloor$ necessarily differ by less than $1$, and $f(x)$ can't be positive, because $\lfloor x\rfloor$ is by definition less than or equal to $x$.

Therefore, the range of $f(x)$ is $\boxed{(-1,0]}$.