Let $k$ be a field and $f(x)\in k[x]$. Let $g(x) = f(\alpha x + \beta)$ for some $\alpha, \beta \in k, \alpha\neq 0$. Prove that $f(x)$ and $g(x)$ have the same discriminants and Galois groups.
I have evaluated the case for when the discriminant is 0, but I'm confused as where to go with the non-zero case.....