We will revisit this concept in Chapter IntroductiontoDerivatives, but for now, we use it as a way to practice function notation and function arithmetic. For reasons which will become clear in Calculus, ‘simplifying’ a difference quotient means rewriting it in a form where the ‘\(h\)’ in the definition of the difference quotient cancels from the denominator. Once that happens, we consider our work to be done.

As mentioned before, we will revisit difference quotients in Section LinearFunctions where we will explain them geometrically. For now, we want to move on to some classic applications of function arithmetic from Economics and for that, we need to think like an entrepreneur.

Suppose you are a manufacturer making a certain product. Let \(x\) be the production level, that is, the number of items produced in a given time period. It is customary to let \(C(x)\) denote the function which calculates the total cost of producing the \(x\) items. The quantity \(C(0)\), which represents the cost of producing no items, is called the fixed cost, and represents the amount of money required to begin production.

Associated with the total cost \(C(x)\) is cost per item, or average cost, denoted \(\overline {C}(x)\) and read ‘\(C\)-bar’ of \(x\). To compute \(\overline {C}(x)\), we take the total cost \(C(x)\) and divide by the number of items produced \(x\) to get

\[ \overline {C}(x) = \frac {C(x)}{x}\]

On the retail end, we have the price \(p\) charged per item. To simplify the dialog and computations in this text, we assume that the number of items sold equals the number of items produced. From a retail perspective, it seems natural to think of the number of items sold, \(x\), as a function of the price charged, \(p\). After all, the retailer can easily adjust the price to sell more product.

In the language of functions, \(x\) would be the dependent variable and \(p\) would be the independent variable or, using function notation, we have a function \(x(p)\). While we adopt this convention elsewhere in the text, we will hold with tradition at this point and consider the price \(p\) as a function of the number of items sold, \(x\). That is, we regard \(x\) as the independent variable and \(p\) as the dependent variable and speak of the price-demand function, \(p(x)\). Hence, \(p(x)\) returns the price charged per item when \(x\) items are produced and sold.

Our next function to consider is the revenue function, \(R(x)\). The function \(R(x)\) computes the amount of money collected as a result of selling \(x\) items. Since \(p(x)\) is the price charged per item, we have \(R(x)= x p(x)\). Finally, the profit function, \(P(x)\) calculates how much money is earned after the costs are paid. That is, \(P(x) = (R-C)(x) = R(x) - C(x)\). We summarize all of these functions below.

Summary of Common Economic Functions

Suppose \(x\) represents the quantity of items produced and sold.

  • The price-demand function \(p(x)\) calculates the price per item.
  • The revenue function \(R(x)\) calculates the total money collected by selling \(x\) items at a price \(p(x)\), \(R(x) = x \, p(x)\).
  • The cost function \(C(x)\) calculates the cost to produce \(x\) items. The value \(C(0)\) is called the fixed cost or start-up cost.
  • The average cost function \(\overline {C}(x) = \frac {C(x)}{x}\) calculates the cost per item when making \(x\) items. Here, we necessarily assume \(x > 0\).
  • The profit function \(P(x)\) calculates the money earned after costs are paid when \(x\) items are produced and sold, \(P(x) = (R-C)(x) = R(x) - C(x)\).

It is high time for an example.