I was recently reading this post and noticed some terminology I am not familiar with. The title of the post is "Why is convex conjugate defined on functions taking values on extended real line?"
What does it mean when we say 'a function $f : X \to \mathbb{R}$ $\cup$ {$-\infty, \infty$} taking values on the extended real line'? Does saying "$f$ is a function taking values on the extended real line (or any set, in a general sense)" mean the function values are elements of $\mathbb{R}$ $\cup$ {$-\infty, \infty$}?
Yes, it is just as you say. A function goes from one set to another. The text is defining what set the range is contained in, which is just as you think $\Bbb R \cup \{-\infty,\infty\}$
The motivation is that there are sets of reals that do not have a supremum in the reals. The set of natural numbers, for example, does not. If we add $\pm \infty$ to the reals, we have a compact set and every subset that is bounded above has a least upper bound. For the naturals, it is $+\infty$.