jmc

We solve the equation $f(x)=0$ on the domains $x\le 3$ and $x>3.$

If $x\le 3,$ then $f(x)=2x+1,$ so we want to solve $2x+1=0.$ The solution is $x=-1/2,$ which satisfies $x\le 3.$

If $x>3,$ then $f(x)=8-4x,$ so we want to solve $8-4x=0.$ The solution is $x=2,$ but this value does not satisfy $x>3.$

Therefore, the only solution is $x=\boxed{-\frac{1}{2}}.$