I want to show that the Legendre Transform of a function $g(x) = kf(x-x_0) + m$ for $m$, $k$ constants is equal to $g^*(p) = kf\left(\frac{p}{k}\right)+p \cdot x_0+m$ where $f^*$ is the Legendre transform of $f$.
I have attempted to solve this:
$$ g^*(p) = \sup_x(p^\top \cdot x - g(x)) = \sup_x(p^\top x - k f(x-x_0)+m) . $$
My question is, can we say that this is equal to $\sup_{x}(p^\top x - kf(x-x_0)+m)$ where the supremum is now taken over all $x-x_0$?
If so, why?