If $a$ and $b$ are elements of a Hopf algebra over a field $k$ and $\alpha, \beta \in k$, then what is $\Delta(\alpha a+\beta b)$?
Is it $\alpha\Delta(a)+\beta\Delta(b)$?
For example if $\Delta(x)=x \otimes 1$, $\Delta(p)=p \otimes 1 + 1 \otimes p$, $\Delta(g) = g \otimes g$ and $\alpha, \beta, \gamma \in k$, then what is $\Delta(\alpha x + \beta p + \gamma g)$?
This property is nowhere to be found in the literature I've looked into (mostly Majid). Also, what would be answer for analogous question for counit?