How to show $Hom(V,V)\rightarrow Hom(V_x,V_x)$ is injective, V being semi-stable

Question

Let $V$ be a semi-stable vector bundle over a smooth irreducible projective curve of genus $g\geq 2$.

Let $x\in X$. How do we show that the canonical map $Hom(V,V)\rightarrow Hom(V_x,V_x)$ which sends $f$ to $f_x$ (i.e., $f$ restricted to the fiber $V_x$) is injective.