Functions are mathematical equations that describe the relationship of a dependent variable to one or more independent variables. Independent variables are exogenous to the functions, meaning that their values change based on the changes of outside variables not included in the functions. In contrast, dependent variables change values based on the changes of independent variables. Production functions are functions that describe the changes in the quantities of products produced due to changes in the resources used in production.
In regard to production functions, the dependent variable is the quantities of the product that is produced. The independent variable or variables are the resources committed to producing that product. In short, the dependent variable is the output, while the independent variables are the inputs. Depending on the specific product and the technologies available, production functions can and will use different independent variables.
Output is the quantities produced of the product. It's an important factor because successful businesses must be able to estimate the optimum quantities of products to produce to sell as much as possible while retaining a price that's profitable to sell at. Once these figures are calculated using other models, production functions may then be used to predict the optimum inputs needed to produce those optimum quantities of products.
Inputs are the resources used to produce the products. Depending on the products, different inputs might be required in the course of production or can be substituted for other inputs. For example, the production of certain assembled goods might be produced using automated machines that can also be substituted for using human labor. Production functions are used to determine the most efficient combination of inputs needed to produce the desired quantities.
Product functions are used in managerial economics to determine the most efficient combination of inputted resources needed to produce a desire amount of products. They're not exact replications of real circumstances and aren't intended to be. Instead, they're abstract models intended to focus on the problem of the efficient usage of resources available to the business.