Consider one variant of conjugate function: $$V(s) = \underset{x}\max \{\langle s,x-x_0\rangle-\beta f(x)\}$$ You can think $s$ as a linear functional.
If I do the following steps:
\begin{equation} \begin{aligned} V(s) &= \underset{x}\max \{\langle s,x\rangle-\beta f(x)\}-sx_0 \\ &= \beta\underset{x}\max\{\langle \frac{1}{\beta}s,x\rangle-f(x)\}-sx_0 \\ &=\beta f^*(\frac{1}{\beta}s)-sx_0 \end{aligned} \end{equation}
Therefore I get the relation between $V(s)$ and $f^*(s)$.
I just want to check if my derivation is correct?