1.2 Implicit Equations

An implicit equation, such as our motivating example

may not be expressed as a function g,

since for x = 0, y=-1 and y=1 both satisfy our equation. All hope is not lost, as our equation may be expressed as the union of two functions:

We may then graph each function separately, and then combine the two graphs into a single graph.

Consider the following equation:

whose graph follows:

If this graph were to be separated into a collection of functions, an infinite number of functions would be needed, since each function may describe at most one point for each value of x. For any value of x, an infinite number of values of y satisfy the equation given. However, for any finite region of the plane, a finite number of functions suffice. Some equations, such as

require an infinite number of functions, even to graph finite regions of the plane, using the procedure just described.

Jeff Tupper

March 1996