Functions

A relation between two variables $x$ and $y$, is any set of points which are on the $(x, y)$ plane.

There exist four different types of relations:

• One to one

• One to many

• Many to one

• Many to many

A function is defined as a mapping onto a single value. Therefore one to one, and many to one relations are considered to be functions.

Testing if a relation is a function

1. Algebraic Method: If we substitute any value of $x$ and it results in a singular $y$-value, then it can be considered a function.
2. Graphical Method (Vertical Line Test): If we are given a plot of a function on the $(x, y)$ plane, where the $x$-axis is parallel with the horizontal, and the $y$-axis is parallel with the vertical, then if we are able to draw a vertical line anywhere on the graph, and it only intersects the plot once, it is a function. If the vertical line intersects the plot more than once, then it cannot be considered a function.

Domain and Range