Relations and Functions in Algebra
Relations
A relation is a rule or a set of rules that connect one set of numbers to another set of numbers. A number or variable from one set can connect to many in the second set and vice versa. The first set of numbers is called the domain while the second set is called the range.
The x and y axes are used for two sets of numbers on a graph. The x axis is the domain while the y axis is the range.
Functions
In a function, a number in the first set can only have one connection to a number in the second set although a number in the second set can have multiple connections from the first set.
Simplest Function
The simplest example of a function is a straight line. To test whether it is a function or not, the vertical line test can be used. If a vertical line crosses a relation on a graph in more that one place then the relation is not a function.
One-To-One Functions
Some functions are one-to-one meaning that for each number in the domain (first set) there is only one number in the range (second set) that corresponds to it.
An example of a one-to-one function is the line or possibly an odd degree polynomial. Another example would be an exponential function or a logarithmic function.
A horizontal line can be used to test whether a function is one-to-one. If the horizontal line doesn't cross the function in more than one place then it is one-to-one. It is important for a function to be defined as one-to-one for it to have an inverse function.
The quadratic function or parabola is not one-to-one since a horizontal line can intersect the function in more than one place. One-to-one functions have inverse functions. An example is the exponential function whose inverse function is the logarithmic function.