Let $f(x)$ be an even function, and let $g(x)$ be an odd function.  Is $f(x) g(x)$ even, odd, or neither?

Enter "odd", "even", or "neither".
Since $f(x)$ is even and $g(x)$ is odd,
\[f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x),\]so $f(x) g(x)$ is an $\boxed{\text{odd}}$ function.