FUNCTION. A mathematical term due to Leibnitz (1692), but first defined in its present sense by Johann Bernoulli (1718). In this sense a function is a quantity whose value depends upon that of another quantity. E.g. in the for mula for the circumference of a circle, c= 2vr, c depends upon r for its value; c is therefore said to be a function of r. Likewise, in the equation the value of y depends upon the value of x, so that if x = .... —2, —1, 0, 1, 2, etc., y = ....3, 2, 3, 6, 11, etc.; y is therefore a function of x, and this is expressed by the symbol y = f(x), which was first employed almost si multaneously by Euler (1734) and Clairaut. In stead of f (x) other symbols are often used, as F (x), (p(X), ifr(x), etc. In y = f (x), the value of y depending upon that of x, x is called the independent and y the dependent variable. In a function like y = ax + b, y is called an ex plicit function of x; in the expression xi + 2xy b = 0, y is an implicit function, and this is indicated by the symbols f (x, y) = O. In the same way we may have f (x, y, a) = 0, f x2, . . . . 0, or we may have z= f (x, y), y = f. If a function has only one value for each given value of the variable, it is called a uniform (monodromic, monotropic, eindeutig) function, as in the case of y = x'+2x+3. But if a function has more than one value for any given value of the variable, or if its value can be changed by modifying the path in which the variable reaches that given value, the func tion is said to be multiform (polytropic, mehr deutig), as in the case of y If the equation y =f (x) be solved for x, then x will equal some function of y, i.e. x = (y), and the latter function is called the inverse of the for mer. E.g. in the case of a sphere v = f (r) = 3 --.
and r = = (v), 4r =f (r) and r = (v) being inverse functions.
Functions were classified by Leibnitz as alge braic or transcendental. The former are such as include only the four fundamental operations to gether with the use of constant exponents, their simplest forms being a + x, ax, x", and their most general form being (a +bx±ce+ • • • •)n (a' + blx . . .
In the broadest sense we say that y is an algebraic function of x when +
. . . . + y + = 0, where A, is a polyno mial in x of the form = x"' + a . . . d-a„,. The transcendental func tions include all other functions, to which, from the domain of the common operations, powers with variable exponents, the so-called exponential functions and their inverse, logarithms, chiefly belong.
An important class of transcendental functions is known as circular functions. These include the goniometric functions, y = sinx, cosx, tanx, eotx, etc. (see TRIGONOMETRY and their in verses, the cyclometric functions, x = or arcsiny, etc. It is shown in trigonometry that y = sin x = sin (x = 2kr), where k is any in teger, so that x may be increased or decreased by 2r,4r, 6r, without altering the value of y; the function is then called simply periodic. In the inverse function, x = x evidently may have, for any value of y, an infinite number of values; this function is therefore called in finitely multiform. The inverse exponential func tion (i.e. the logarithm) and the circular func tion are integrals of algebraic functions. Thus, Cdx =log x,f dx = sin J v 1 --Le dx—tan c J —1 =see etc., all with the proper constants.
If a function y = f(x), or co (x, y)= 0, be plotted, the figure is a curve with infinitely many points in immediate succession. The continuity of the curve and, corresponding to it, the con tinuity of the function, consist in this, that any two successive points lie infinitely near each other, so that an infinitely small variation of the abscissas is attended by an infinitely small varia tion of the ordinates. This suffices to explain what is meant by a continuous function, the meaning of the term discontinuous function be ing easily inferred. E.g. the functions a + ax, ax, sinx, coax, are continuous in the domain (- ± OD) of the variable x, as is also xn when n is a positive integer. The functions -,/x, logx are continuous in the domain (0, + cc), The function --, where n is a positive integer, x' is continuous in the domains ( —cc, o — e), (o + e, + cc), however small e may be; but for x = 0 it breaks its continuity and y = =co.