2019 AMC 10B Problem 9

If $x\ge0$, then $\lfloor|x|\rfloor=\lfloor x\rfloor=|\lfloor x\rfloor|$, so $f(x)=0$. If $x$ is a negative integer, both terms equal $|x|$, so again $f(x)=0$. If $x$ is negative and not an integer, write $x=-k-t$, where $k$ is a nonnegative integer and $0<t<1$. Then $\lfloor|x|\rfloor=k$, while $|\lfloor x\rfloor|=|-k-1|=k+1$, so $f(x)=-1$. Therefore, the range is $\{-1,0\}.$ Thus, the answer is A .