2

y i1

Figure 2.7

where the arrow indicates mapping, and the letter/ symbolically specifies a rule of mapping. Since / represents a particular rule of mapping, a different functional notation must be employed to denote another function that may appear in the same model. The customary symbols (besides/) used for this purpose are g, F, G. the Greek letters <j> (phi) and ip (psi), and their capitals, $ and For instance, two variables y and z may both be functions of x, but if one function is written as y = f(x), the other should be written as z = g(x), or z = <f>(x). It is also permissible, however, to writey = y(x) and z = z(x), thereby dispensing with the symbols / and g entirely.

In the function y = f(x), x is referred to as the argument of the function, and v is called the value of the function. We shall also alternatively refer to x as the independent variable and y as the dependent variable. The set of all permissible values that x can take in a given context is known as the domain of the function, which may be a subset of the set of all real numbers. The y value into which an a value is mapped is called the image of that x value. The set of all images is called the range of the function, which is the set of all values that the y variable will take. Thus the domain pertains to the independent variable x, and the range has to do with the dependent variable y.

As illustrated in Fig. 2.7a, we may regard the function/ as a rule for mapping each point on some line segment (the domain) into some point on another line segment (the range). By placing the domain on the x axis and the range on the>' axis, as in diagram b, however, we immediately obtain the familiar two-dimensional graph, in which the association between x values and y values is specified by a set of ordered pairs such as (x,, yt) and (x2, y2)-

In economic models, behavioral equations usually enter as functions. Since most variables in economic models are by their nature restricted to being nonnegative real numbers,* their domains are also so restricted. This is why most

* We say "nonnegative" rather than "positive" when zero values are permissible.

geometric representations in economics are drawn only in the first quadrant. In general, we shall not bother to specify the domain of every function in every economic model. When no specification is given, it is to be understood that the domain (and the range) will only include numbers for which a function makes economic sense.

Example 5 The total cost C of a firm per day is a function of its daily output Q: C = 150 + 1Q. The firm has a capacity limit of 100 units of output per day. What are the domain and the range of the cost function? Inasmuch as Q can vary only between 0 and 100, the domain is the set of values 0 < Q < 100; or more formally,

As for the range, since the function plots as a straight line, with the minimum C value at 150 (when Q = 0) and the maximum C value at 850 (when Q = 100), we have

Beware, however, that the extreme values of the range may not always occur where the extreme values of the domain are attained.

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