Function.

Function. When two quantities are so related that a change in the one produces a corresponding change in the other, the latter is termed a function of the former. For example, the area of a triangle is a function of the base, since the area decreases or increases with the decrease or increase of the base, the altitude remaining unchanged. Again, if $u\; =\; ax^2\; +\; bx\; +\; c$, where $a$, $b$, and $c$ are constant quantities, and $u$ and $x$ variables; then $u$ is said to be a function of $x$, since, by assigning to $x$ a series of different values, a corresponding series of values of $u$ is obtained, showing its dependence on the value given to $x$. Moreover, for this reason, $x$ is termed the independent, $u$ the dependent variable. There may be more than one independent variable—e.g. the area of a triangle depends on its altitude and its base, and is thus a function of two variables. Functionality, in algebra, is denoted by the letters $F$, $f$, $\backslash phi$, $\backslash Phi$, &c. Thus, that $u$ is a function of $x$ may be denoted by the equation $u\; =\; F(x)$; or, if the value of $u$ depends on more than one variable, say upon $x$, $y$, and $z$, then by $u\; =\; F(x,\; y,\; z)$.

Functions are primarily classified as algebraical or transcendental. The former include only those functions which may be expressed in a finite number of terms, involving only the elementary algebraical operations of addition, subtraction, multiplication, division, and root extraction. Several terms are employed to denote the particular nature of such functions. A rational function is one in which there are no fractional powers of the variable or variables; integral functions do not include the operation of division in any of their terms; a homogeneous function is one in which the terms are all of the same degree—i.e. the sum of the indices of the variables in each term is the same for every term. For example,

is a rational, integral, homogeneous function of the fourth degree in $x$ and $y$. Transcendental functions are those which cannot be expressed in a finite number of terms; the principal types are (1) the exponential function $e^x$, and its inverse, $\backslash log\; x$; (2) the circular functions, such as $\backslash sin\; x$, $\backslash cos\; x$, $\backslash tan\; x$, &c., and their respective inverses, $\backslash sin^\{-1\}x$, $\backslash cos^\{-1\}x$, $\backslash tan^\{-1\}x$, &c.

Functions are also distinguished as continuous or discontinuous. Any function is said to be continuous when an infinitely small change in the value of the independent variable produces only an infinitely small change in the dependent variable; and to be discontinuous when an infinitely small change in the independent variable makes a change in the dependent variable either finite or infinitely great. All purely algebraic expressions are continuous functions; as are also such transcendental functions as $e^x$, $\backslash log\; x$, $\backslash sin\; x$, $\backslash cos\; x$.

Harmonic or periodic functions are those whose values fluctuate regularly between certain assigned limits, passing through all their possible values, while the independent variable changes by a certain amount known as the period. Such functions are of great importance in the theory of sound, as well as in many other branches of mathematical physics. Their essential feature is that, if $f(x)$ be a periodic function whose period is $a$, then $f(x\; +\; \backslash frac\{1\}\{2\}a)\; =\; f(x\; -\; \backslash frac\{1\}\{2\}a)$, for all values of $x$.

The term derived function is used to denote the successive coefficients of the powers of $h$ in the expansion of $f(x+h)$, where $h$ is an increment of $x$. If $x$ becomes $x+h$, then $f(x)$ changes to $f(x+h)$, and it may be shown that ; $f\text{'}(x)$, $f\text{'}\text{'}(x)$, $f\text{'}\text{'}\text{'}(x)$, &c. are the first, second, third, &c. derived functions of $f(x)$. It is the primary object of the differential calculus to find the value of these for different kinds of functions.