Mathematical Modeling in the Social and Life Sciences

APPENDIX IV

Functions of Two Variables

Let S be a subset of the (x, y)-plane. A relationship that assigns a unique number to each point of S is called a real-valued function of two variables. The domain of such a function is a set of ordered pairs (x, y) of real numbers, and the range is a subset of the reals. We may denote such a function by the letters customarily reserved for functions: f, g, h, F, G, H, ϕ, θ, . . . .

We write

to denote that f assigns the number z to the ordered pair (x, y).

EXAMPLE 1

Consider the function f(x, y) = x² + y⁴. We then have f(9, 2) = 9² + 2⁴ = 81 + 16 = 97, f(−9, 2) = 97, and f(−7, 0) = 49. This function is defined for all values of x and y, so its domain is the entire plane. Since x² + y⁴ is the sum of two nonnegative numbers, no negative numbers can be in the range. On the other hand, if z is any nonnegative real number, then f(z^1/2, 0) = z. Thus, the range of f is the set of all nonnegative real numbers. Since f(9, 2) = f(−9, 2) = f(9, −2) = f(−9, −2), the function is not one-to-one.

EXAMPLE 2

Let f be the function given by f(x, y) = Then we have f(4, 3) = 1/6, f(9, −1/3) = −1, while f(−5, 2) and f(2, 0) are undefined. Now f is defined whenever x is positive and y is nonzero. Thus, the domain of f is the open right half-plane excluding the ...