Functions Of Two Or More Independent Variables

Thus far, we have considered only functions of a single independent variable, y = f(x). But the concept of a function can be readily extended to the case of two or more independent variables. Given a function

a given pair of x and y values will uniquely determine a value of the dependent variable z. Such a function is exemplified by z = ax + by or z = a0 + a^x + a2x2 + bty + b2y2

Just as the functiony — f(x) maps a point in the domain into a point in the range, the function g will do precisely the same. However, the domain is in this case no longer a set of numbers but a set of ordered pairs (x, > ), because we can determine z only when both x and y are specified. The function g is thus a mapping from a point in a two-dimensional space into a point on a line segment

(i.e., a point in a one-dimensional space), such as from the point (x,, y}) into the point z, or from (x2, y2) into z2 in Fig. 2.9a.

If a vertical z axis is erected perpendicular to the xy plane, as is done in diagram b, however, there will result a three-dimensional space in which the function g can be given a graphical representation as follows. The domain of the function will be some subset of the points in the xy plane, and the value of the function (value of z) for a given point in the domain—say, (xt, >',)—can be indicated by the height of a vertical line planted on that point. The association between the three variables is thus summarized by the ordered triple (x,, >',, zt), which is a specific point in the three-dimensional space. The locus of such ordered triples, which will take the form of a surface, then constitutes the graph of the function g. Whereas the function y = /(x) is a set of ordered pairs, the function

z = g(x, y) will be a set of ordered triples. We shall have many occasions to use functions of this type in economic models. One ready application is in the area of production functions. Suppose that output is determined by the amounts of capital (K) and labor (L) employed; then we can write a production function in the general form q = q(k, l).

The possibility of further extension to the cases of three or more independent variables is now self-evident. With the function y = h(u, v, w), for example, we can map a point in the three-dimensional space, («,, u^vv,), into a point in a one-dimensional space (j^,). Such a function might be used to indicate that a consumer's utility is a function of his consumption of three different commodities, and the mapping is from a three-dimensional commodity space into a one-dimensional utility space. But this time it will be physically impossible to graph the function, because for that task a four-dimensional diagram is needed to picture the ordered quadruples, but the world in which we live is only three-dimensional. Nonetheless, in view of the intuitive appeal of geometric analogy, we can continue to refer to an ordered quadruple («,, r,. w,, y,) as a "point" in the four-dimensional space. The locus of such points will give the (nongraphable) graph of the function y = h(u, t\ w), which is called a hypersurface. These terms, viz., point and hypersurface, are also carried over to the general case of the n-dimensional space.

Functions of more than one variable can be classified into various types, too. For instance, a function of the form

is a linear function, whose characteristic is that every variable is raised to the first power only. A quadratic function, on the other hand, involves first and second powers of one or more independent variables, but the sum of exponents of the variables appearing in any single term must not exceed two.

Note that instead of denoting the independent variables by x, u, v, w, etc., we have switched to the symbols x,, x2_____ xn. The latter notation, like the system of subscripted coefficients, has the merit of economy of alphabet, as well as of an easier accounting of the number of variables involved in a function.

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