FUNCTION

in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). In its most general usage in mathematics the word function refers to any correspondence between two classes. For most functions the variables range over classes of numbers. For example, the formula A = pr2 gives for each positive real number r the area of the circle with radius r. The expressions a + bx + cx2 and a0 + a1x + . . . + anxn are polynomial functions of x when the coefficients a, b, c, a0, a1, . . . , an are given. The short symbols f (x), g(x), P(x), . . . , are often used for functions of the independent variable x, either for the sake of abbreviation, or because the nature of the function is unknown or unspecified. The quotient of two polynomials P(x)/Q(x) is called a rational function. A polynomial is regarded as a special case of a rational function. The trigonometric functions sin x, cos x, tan x and others, where x is the measure of an angle, are defined geometrically in elementary trigonometry; for practical purposes their values are given in tables. Many functions of practical importance are defined only by means of tables, as in statistics. For example, the death rate in each year over a period of years is given in mortality tables used by life insurance companies. Inverse functions are obtained from given functions by interchanging the roles of the independent and dependent variables. Thus, if the given function is written y = 2x, the inverse function would be written x = y/2. The two functions determine the same correspondence between the two variables. The exponential function y = 10x gives a value of y for each real value of x. These values are not easily computed unless x is an integer. In this case the inverse function is written x = log10y and its values are given in tables of common logarithms. Functions involving more than two variables occur frequently in applications of mathematics. For example, the formula A = 1/2bh gives the area of a triangle in terms of its base b and altitude h.