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.