Relationships – or relations – express the ways in which two quantities are related. They can be represented as sets of ordered pairs, tables of values, mapping diagrams, rules, or graphs.
When a relationship is represented with a rule, the rule describes a mathematical relationship between the input and the output of the function.
Special types of relations, called functions, are presented as a relation in which each input value corresponds to exactly one output value. There are different ways to determine if a relation is a function depending on the way the relation is represented, including the use of mapping diagrams and the Vertical Line Test.
The set of all input values for which a function is defined is called the domain of the function. The set of all output values is referred to as the function's range.
Function notation, f(x)=y, is introduced to show how one quantity is dependent upon another. Because functions show how two quantities relate, they are useful in analyzing situations in context.
Key features of the graphs of functions are used to further describe the behavior of functions. Commonly used key features include x- and y-intercepts, increasing and decreasing intervals, relative minimum and maximum values, symmetries and end behavior.
Many kinds of functions have been given names. Some of the more commonly used functions are linear functions, polynomial functions, and exponential functions.