Since any ordered pair associates ay value with an x value, any collection of ordered pairs—any subset of the Cartesian product (2.3)—will constitute a relation between y and .v. Given an x value, one or more y values will be specified by that relation. For convenience, wc shall now write the elements of* x y generally as (x, y) rather than as (a, b), as was done in (2.3) where both x andy are variables.
The set ((*, y) | y — 2a] is a set of ordered pairs including, for example, (1, 2), (0, 0), and (-1, -2). It constitutes a relation, and its graphical counterpart is the set of points lying on the straight line y = 2xf as seen in Fig. 2.5.
FIGURE 2.5
FIGURE 2.5
4~~ The set {(x, y) | y < x], which consists of such ordered pairs as (1, 0), (1y 1), and (1, -4), — constitutes another relation. In Fig. 2.5, this set corresponds to the set of alf points in the shaded area which satisfy the inequality y < x.
4~~ The set {(x, y) | y < x], which consists of such ordered pairs as (1, 0), (1y 1), and (1, -4), — constitutes another relation. In Fig. 2.5, this set corresponds to the set of alf points in the shaded area which satisfy the inequality y < x.
Observe that when the x value is given, it may not always be possible to determine a uniquey value from a relation. In Example 4, the three exemplary ordered pairs show that if x — 1, y can take various values, such as 0, 1, or —4, and yet in ouch case satisfy the stated relation. Graphically, two or more points of a relation may fall on a single vertical line in the xy plane. This is exemplified in Fig. 2.5, where many points in the shaded area (representing the relation v < x) fall on the broken vertical line labeled x = a.
As a special case, however, a relation may be such that for each x value there exists only one corresponding^ value. The relation in Example 3 is a case in point. In such a case, y is said to be & function of x? and this is denoted by y = f(x), which is read as "v equals/of x.77 [Nate: f(x) does not mean/times x.1 A function is therefore a set of ordered pairs with the properly that any x value uniquely determines a v value.1 It should be clear that a function must be a relation, but a relation may not be a function.
Although the definition of a function stipulates a unique y for each a . the converse is not required. In other words, more than one value may legitimately be associated with the same y value. This possibility is illustrated in Fig. 2,6, where the values a-i and x2 m the x set are both associated with the same value (Jq) in they set by the function y — fix).
A function is also called a mapping, or transformation; both words connote the action of associating one thing with another. In the statement y = f{x ), the functional notation /
T This definition of function corresponds to wh^t wguld be called a single-valued function in the older terminology. What was formerly cafled a multivalued function h now referred to as a relation or correspondence.
may thus be interpreted to mean a rule by which the set x is "mapped" ("transformed") into the set y. Thus we may write f\x y where the arrow indicates mapping, and the letter f 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 $ (phi) and f (psi), and their capitals, <t> and For instance, two variables >> and 2 may both be functions of x. but if one function is written as y = /(x), the other should be written as z — g(x), or z — It is also permissible, however, to write y = y(x) and z = z(x}, thereby dispensing with the symbols/and g altogether.
In the function y =±= /(x), x is referred to as the argument of the function, andy is called the value of the function. We shall also alternatively refer to x as the independent variable and>f 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 v value into which an x 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 can 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 y axis, as in Fig. 2.7/?, 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 (xu y\ ) and (X2, yi).
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,1 their domains are also so restricted. This is why most geometric representations in
* We say ''nonnegative" rather than "positive" when zero values are permissible.
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) wall only include numbers for which a junction makes economic sense.
The total cost C of a firm per day is a function of its daily output Q: C = 150 + 7Q. 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 H 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.
EXERCISE 2.4
1, Given Si = 13,6,9}t S2 - {a, b\, and S3 = (m, n], find the Cartesian products: (0) Si x S2 (b) S2 x S3 (c) x Si
2. From the information in Prob. 1, find the Cartesian product St x $2 x S3.
In general, is it true that Si x S2 = S2 x Si? Under what conditions will these two Cartesian products be equal?
4. Does any of the following, drawn in a rectangular coordinate plane, represent a function?
{b) A triangle (d) A downward-sloping straight line
5. If the domain of the function y ™ 5 -+- 3x is the set [x | 1 < x < 9), find the range of the function and express it as a set.
FIGURE 2.7
Example 5
6. For the function y = -x2, if the domain is the set of all nonnegative real numbers, what will its range be?
7. In the theory of the firm, economists consider the total cost C to be a function of the output level Q: C = f(Q).
(o) According to the definition of a function, should each cost figure be associated with a unique level of output? (fa) Should each level of output determine a unique cost figure?
8. If an output level Qi can be produced at a cost of ti, then it must also be possible (by being less efficient) to produce Qi at a cost of C} + $1, or Q + $2, and so on. Thus it would seem that output Q does not uniquely determine total cost C. If so, to write C - f(Q) would violate the definition of a function. How, in spite of the this reasoning, woutd you justify the use of the function C = f( Q)7
The expression y = f{x) is a general statement to the ellecl that a mapping is possible, but the actual rule of mapping is not thereby made explicit. Now let us consider several specific types of Junction, each representing w different rule of mapping.
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