The production function simply states the quantity of output (Q) that a firm can produce as a function of the quantity of inputs to production, or there can be a number of different inputs to production, i.e. “factors of production.” but they are generally designated as either capital or labor. (Technically, land is a third category of factors of production, but it’s not generally included in the production function except in the context of a land-intensive business.) The particular functional form of the production function (i.e. the specific definition of I) depends on the specific technology and production processes that a firm uses.
In the short run, the amount of capital that a factory uses is generally thought to be fixed. (The reasoning is that firms must commit to a particular size of factory, office, etc. and can’t easily change these decisions without a long planning period.) Therefore, the quantity of labor (L) is the only input in the short-run production function.
In the long run, on the other hand, a firm has the planning horizon necessary to change not only the number of workers but the amount of capital as well, since it can move to a different size factory, office, etc. Therefore, the long-run production function has two inputs that be changed- capital (C) and labor (L). Both cases are shown in the diagram above.
Note that the quantity of labor can take on a number of different units- worker-hours, worker-days, etc. The amount of capital is somewhat ambiguous in terms of units, since not all capital is equivalent, and no one wants to count a hammer the same as a forklift, for example. Therefore, the units that are appropriate for the quantity of capital will depend on the specific business and production function