Implicit Curve Definition

An implicitly defined curve is a curve that is described by an equation involving two or more variables, rather than by an explicit function in terms of one variable. The equation defines the relationship between the variables but does not explicitly provide one variable as a function of the other.

For example, the equation defines a circle with radius centered at the origin. This equation describes the relationship between and , but does not provide an explicit function for in terms of , or vice versa.

The equation defines the unit hyperbola.

The equation defines the ellipse with a major axis of and a minor axis of .

Implicitly defined curves are analyzed using techniques like implicit differentiation to find properties such as tangent lines and slopes.

Function Definition

A function is a mathematical relationship between two sets, called the domain and the codomain, in which each element in the domain corresponds to exactly one element in the codomain.