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Real Function of Two Variables
DEFINITION A real function of two variables is a set f of ordered triples of real numbers such that for every ordered pair of real numbers (a, b) one of the following two things occurs: (i) There is exactly one real number c for which the ordered triple (a, b, c) is a member of f. In this case, f(a, b) is defined and we write: f(a, b) = c (ii) There is no real number c for which the ordered triple (a, b, c) is a member of f. In this case f(a, b) is called undefined. If f is a real function of two variables, then the value of f(x,y) depends on both the value of x and the value of y when f(x,y) is defined. A real function f of two variables can be visualized as a black box with two input lines and one output line, as in Figure 1.2.12.
Figure 1.2.12 The domain of a real function f of two variables is the set of all pairs of real numbers (x, y) such that f(x, y) is defined. The most important examples of real functions of two variables are the sum, difference, product, and quotient functions: f(x, y) = x + y, f(x, y) = x · y, f(x, y) = x - y, f(x, y) = x / y. The sum, difference, and product functions have the whole plane as domain. The domain of the quotient function is the set of all ordered pairs (x, y) such that y ≠ 0.
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Last Update: 2006-11-09