Algebraic Functions

A function is a rule, correspondence, or mapping x \mapsto y that assigns to each real number x (the input) to a certain set D 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 and is called the range of the function.

Most calculators have special keys for some of the more important functions such as x \mapsto \sqrt x and x \mapsto {x^2}. 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 x \mapsto \sqrt x by the letter f, we write f:x \mapsto \sqrt x .

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

For instance, if f:x \mapsto \sqrt x is the square-root function, then f\left( 4 \right) = \sqrt 4 = 2, f\left( {25} \right) = \sqrt {25} = 5, f\left( 2 \right) = \sqrt 2 \approx 1.414 etc, and in general, for any nonnegative value of x, f\left( x \right) = \sqrt x . Note carefully that f\left( x \right) does not mean f times x.