Now it's time to throw a monkey wrench into the works, curve ball style. What happens when we mix two variables together, on both sides of the equation? Why calculus? Why?! We're entering the Twilight Zone; we're traveling to another dimension. Once the room stops spinning, we'll move on. Good now? Good. We can handle this.
A function is a special kind of relation where each input has exactly one output. Often we have a rule or a formula that tells us how to reach the input to the output, such as
y = x + 5.
In such cases we say the function is defined explicitly. The dependent variable appears all by itself on one side of the equation, and the other side gives a recipe for making the dependent variable from the independent variable.
We say a function is defined implicitly if we have an equation in which the dependent variable doesn't have a side of the equation all to itself. The equation
5x2 – 4y = 7
is an implicit definition of y, since it doesn't actually say how to go from x to y. If we solve this equation for y we find the explicit definition
We can say that y is "defined implicitly" even if y isn't actually a function. For example, the equation
x2 + y2 = 1
describes a circle of radius 1. A circle isn't a function, since it fails the vertical line test. However, we would still say that this equation implicitly defines y.
For a total headache, we might even say "y is an implicitly defined function" even though y isn't a function. We probably won't worry about this distinction in class, but we wanted to point it out for the sake of completeness.