A function is a rule, correspondence, or mapping that assigns to each real number x (the input) in a certain set D one and one real number y (the output). The set D is called the domain of the function, and y is the dependent variable, since its value depends on the value of x. Because x can assign any value in the domain D, we refer to x as the independent variable. The set of values assumed by y as x runs through all values in D is called the range of the function.

            Most calculators have special keys for some of the more important functions such as  and. The use of letters of the alphabet to designate functions is not restricted exclusively to calculating machines. Although any letters of the alphabet can be us designate functions, the letters f, g, and h as well as F, G, and H are most common. (Letters of the Greek alphabet are also used.) For instance, if we wish to d the square-root function by the letter f, we write  or.

            If  is a function, it is customary to write the value of y that corresponds to x as f(x), read “ f of x” In other words, f(x) is the output produced when function f is applied to the input x.

For instance, if  is the square-root function, then, ,  etc, and in general, for any nonnegative value of x, . Note carefully that f(x) does not mean f times x.