If $\alpha$ and $\beta$ are constants, and $g$ and $h$ are functions such that $g(X)$ and $h(X)$ are random variables.
I want to prove this with discrete variables. $$E(\alpha g(X)+\beta h(X))=\alpha E(g(X))+\beta E(h(X))$$
So I don't know if this is right: $$\alpha E(g(X))+\beta E(h(X))= \alpha \sum_i g(x_i)P(X=x_i)+\beta \sum_i h(x_i)P(X=x_i)$$ $$=\sum_i\alpha g(x_i)P(X=x_i)+\sum_i \beta h(x_i)P(X=x_i)$$ $$=\sum_i (\alpha g(x_i)+\beta h(x_i))P(X=x_i)$$ $$=E(\alpha g(X)+\beta h(X))$$