How will be the Graph of the function $f(x,y)$ where $y=ϕ(x)$. From my intuition it seems it will not be a surface rather than a 3D curve! But I am not sure!

What is the relation between two different functions, say $g(x)$ and $f(x)$, which have the same Legendre–Fenchel transformation $h(s)$? \begin{equation} h(s) = \sup_{x\in I}\{sx - f(x)\} \quad \quad x \in I \end{equation} \begin{equation} h(s) = \sup_{x\in I'}\{sx - g(x)\} \quad \quad ...