A

**function** is defined as a set of ordered pairs (

*x,y*), such that for each first element

*x*, there corresponds one and only one second element

*y*. The set of first elements is called the

*domain* of the function, while the set of second elements is called the

*range* of the function. The domain variable is referred to as the independent variable, and the range variable is referred to as the dependent variable. The notation

*f*(

*x*) is often used in place of

*y* to indicate the value of the function

*f* for a specific replacement for

*x* and is read “

*f* of

*x*” or “

*f* at

*x*.”

Geometrically, the graph of a set or ordered pairs ( *x,y*) represents a function if any vertical line intersects the graph in, at most, one point. If a vertical line were to intersect the graph at two or more points, the set would have one *x* value corresponding to two or more *y* values, which clearly contradicts the definition of a function. Many of the key concepts and theorems of calculus are directly related to functions.

**Example 1:** The following are some examples of equations that are functions.

**Example 2:** The following are some equations that are not functions; each has an example to illustrate why it is not a function.