Step By Step Calculus » 11.1 - Two Variable Functions and Implicitly Defined Curves

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Synopsis
A real-valued function of two variables z=f(x,y)\;:\; D\to Ez=f(x,y)\;:\; D\to E is a mapping that assigns each pair (x, y)(x, y) in D\subseteq\mathbb{R}^2D\subseteq\mathbb{R}^2 to a value zz in E\subseteq\mathbb{R}E\subseteq\mathbb{R} where D, ED, E are the domain and the range of the function ff respectively.
Given a real number cc, the corresponding Level Set of a given function f(x,y)f(x,y) is the set of all pairs (x,y)(x,y) for which f(x,y) = cf(x,y) = c. That is, the level set = L_f(c)=\{(x,y)\;|\; f(x,y)=c\} = L_f(c)=\{(x,y)\;|\; f(x,y)=c\} for a given constant c.c.
So far, we have largely dealt with functions and curves represented by explicit equations of the form y=f(x)y=f(x) or x=f(y)x=f(y). In this section, we study a more general new method for representing functions and plane curves. A function or curve can be represented by an implicit expression. A function or curve is said to be implicitly defined if equation must be solved to find an explicit expression for calculating its values. A simple example is the implicit formula x^{2} + y^{2}= 1, \; y \geq 0x^{2} + y^{2}= 1, \; y \geq 0 which defines the upper half of the unit circle. The explicit formula for this curve (which is the graph of the corresponding implicitly defined function) would be y = \sqrt{1-x^{2}} y = \sqrt{1-x^{2}}.
A curve is closed if it has no endpoints, and completely encloses a certain region in the plane.