I'm stuck on the following exercise, which seems rather simple:
Let $f: (X,\mathcal{O}_X)\longrightarrow(Y,\mathcal{O}_Y)$ be a morphism of varieties and assume that the corresponding morphism of $k$-algebras $f^{\ast}:\ \mathcal{O}_Y(Y)\longrightarrow\mathcal{O}_X(X)$ is injective. Show that $fX$ is dense in $Y$.
A hint would be very welcome.