The graph of a function $f$ passes through the point $(3,-2)$ and has a $y$-intercept of $(0,5)$. Another function $g$ is defined by $g(x)=f(x)+k$, where $k$ is a constant. If the graph of $g$ passes through the point $(3,1)$, what is the $y$-intercept of the graph of $g$?