13. Let real numbers $x, y$ satisfy the constraint conditions $\left\{\begin{array}{l}x+2 y \geqslant 2, \\ 2 x+y \leqslant 4, \\ 4 x-y \geqslant 1 .\end{array}\right.$

Then the range of the objective function $z=3 x+y$ is
13. $\left[\frac{19}{9}, 6\right]$.
According to the problem, it is known that the constraint conditions form a triangle with three vertices at $(2,0)$, $\left(\frac{4}{9}, \frac{7}{9}\right)$, and $\left(\frac{5}{6}, \frac{7}{3}\right)$.

Calculating, we get $z \in\left[\frac{19}{9}, 6\right]$.