Examples of how economists come up with a production function for a firm? I am a young man studying economics on his own. In every microeconomics text I find, they teach you what a production function is, but they never show examples (or practice problems) on how one comes up with a production function of a firm on his/her own. Even in practice problems, a production function is already given. I can never find practice problems that help you practice finding an actual production function (ex: A firm has 3 ovens, 2 computers, 10 special extractors, and 15 workers in an olive oil plant. Find the production function for this firm).

I personally want to see how an economist makes up his/her own production function of a firm he/she is observing. I know this takes some sort of mathematical ability. I personally am not that good at modeling real life situations into functions yet (I am practicing that as well). This is why I want to practice this particular skill. 

Can anyone please show me examples of this? Does anyone have any good websites or books that show great examples of this, along with practice problems? I have found nothing in almost every book/website I've found. Please help. Thank you.
The "production function" in economics and econometrics is not an engineering concept. It is a highly abstract construction where the myriads of inputs used are lumped together in very few categories: "capital" and "labor" being the most usual ones, while sometimes inputs critical to an industry like say "Energy" may be singled out and appear in the production-function formulation. At such an abstract level, it is easily implementable without the need to go into particulars of each firm.

Moreover, in many if not most of the cases, what is modeled is the production of value added, not the total output (either in quantities or in values).  

Applied work commissioned by specific (usually very large) firms, and also, applied research into "adjustment costs" or what is called "insider econometrics", may go some steps towards creating much more complex production functions, that essentially try to take into account the actual physical relations and interactions between the various inputs. But these are very specialized areas -remember that the Economics discipline ultimately is about markets and economic systems -the "individual" (firm or person) level is researched mainly in order to have sound foundations for the aggregation that follows.