Suppose the function $f(x)$ is defined explicitly by the table$$\begin{array}{c || c | c | c | c | c} x & 0 & 1 & 2 & 3 & 4 \\
\hline
f(x) & 0 & 0 & 1 & 3 & 6
\end{array}$$This function is defined only for the values of $x$ listed in the table.

Suppose $g(x)$ is defined as $f(x)-x$ for all numbers $x$ in the domain of $f.$ How many distinct numbers are in the range of $g(x)?$