The graph of a polynomial function $g$ has x-intercepts at $(0,0)$, $(3,0)$, and $(7,0)$. Function $f$ is defined such that $g(x)=f(x-2)$. The graph of $y=f(x)$ in the $xy$-plane has x-intercepts at $(r,0),(s,0)$, and $(t,0)$, where $r,s$, and $t$ are distinct constants. What is the value of $r+s+t$?