Let $f: G\to H$ be a homomorphism of linear algebraic groups. Let $f^*: k[H]\to k[G]$ be the corresponding Hopf algebra morphism. Then $f^*$ factors as $k[H] \twoheadrightarrow k[H]/I\hookrightarrow k[G] $, where $I=\ker f^*$. This means that $f: G\to G'\hookrightarrow H$ where $k[G']\cong k[H]/I$.
Is $G'$ the image of $f$ here? I know that $G\to G'$ is a quotient map (since $k[G']\hookrightarrow k[G]$ is injective), but what does this sat about the image of $f$ specifically?