function

/fungk"sheuhn/, n.

1. the kind of action or activity proper to a person, thing, or institution; the purpose for which something is designed or exists; role.

2. any ceremonious public or social gathering or occasion.

3. a factor related to or dependent upon other factors: Price is a function of supply and demand.

4. Math.

a. Also called correspondence, map, mapping, transformation. a relation between two sets in which one element of the second set is assigned to each element of the first set, as the expression y = x²; operator.

b. Also called multiple-value function. a relation between two sets in which two or more elements of the second set are assigned to each element of the first set, as y² = x², which assigns to every x the two values y = +x and y = -x.

c. a set of ordered pairs in which none of the first elements of the pairs appears twice.

5. Geom.

a. a formula expressing a relation between the angles of a triangle and its sides, as sine or cosine.

b. See hyperbolic function.

6. Gram.

a. the grammatical role a linguistic form has or the position it occupies in a particular construction.

b. the grammatical roles or the positions of a linguistic form or form class collectively.

7. Sociol. the contribution made by a sociocultural phenomenon to an ongoing social system.

v.i.

8. to perform a specified action or activity; work; operate: The computer isn't functioning now. He rarely functions before noon.

9. to have or exercise a function; serve: In earlier English the present tense often functioned as a future. This orange crate can function as a chair.

[1525-35; < L function- (s. of functio) a performance, execution, equiv. to funct(us) (ptp. of fungi) performed, executed + -ion- -ION]

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I

In mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another (the dependent variable), which changes along with it.

Most functions are numerical; that is, a numerical input value is associated with a single numerical output value. The formula A = πr², for example, assigns to each positive real number r the area A of a circle with a radius of that length. The symbols f(x) and g(x) are typically used for functions of the independent variable x. A multivariable function such as w = f(x, y) is a rule for deriving a single numerical value from more than one input value. A periodic function repeats values over fixed intervals. If f(x + k) = f(x) for any value of x, f is a periodic function with a period of length k (a constant). The trigonometric functions are periodic. See also density function; exponential function; hyperbolic function; inverse function; transcendental function.

II

(as used in expressions)

density function

exponential function

hyperbolic function

inverse function

production function

propositional function

transcendental function

trigonometric function

wave function

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▪ mathematics

Introduction

      in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet (Dirichlet, Peter Gustav Lejeune):

If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.

      This relationship is commonly symbolized as y = f(x). In addition to f(x), other abbreviated symbols such as g(x) and P(x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified.

Common functions

      Many widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = πr², gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables also are common in mathematics, as can be seen in the formula for the area of a triangle, A = bh/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values—thus, any real number—the functions are known as real-valued functions.

      The formula for the area of a circle is an example of a polynomial function. The general form for such functions is

P(x) = a₀ + a₁x + a₂x²+⋯+ a_nxⁿ,

where the coefficients (a₀, a₁, a₂,…, a_n) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,…). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility—practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five—linear, quadratic, cubic, quartic, and quintic.

 Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). The graph of the function then consists of the points with coordinates (x, y) where y = f(x). For example, the graph of the cubic equation f(x) = x³ − 3x + 2 is shown in the figure—>.

 Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle (see figure—>). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or “cycles.” Nonalgebraic functions, such as exponential (exponential function) and trigonometric functions, are also known as transcendental functions.

Complex functions

 Practical applications of functions whose variables are complex numbers (complex number) are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + iy, where i is the imaginary unit (the square root of −1) and x and y are real variables (see figure—>), it is possible to split the complex function into real and imaginary parts: f(z) = P(x, y) + iQ(x, y).

Inverse functions

      By interchanging the roles of the independent and dependent variables in a given function, one can obtain an inverse function. Inverse functions do what their name implies: they undo the action of a function to return a variable to its original state. Thus, if for a given function f(x) there exists a function g(y) such that g(f(x)) = x and f(g(y)) = y, then g is called the inverse function of f and given the notation f⁻¹, where by convention the variables are interchanged. For example, the function f(x) = 2x has the inverse function f⁻¹(x) = x/2.

Other functional expressions

      A function may be defined by means of a power series. For example, the infinite series

could be used to define these functions for all complex values of x. Other types of series and also infinite products may be used when convenient. An important case is the Fourier series, expressing a function in terms of sines and cosines:

      Such representations are of great importance in physics, particularly in the study of wave motion and other oscillatory phenomena.

      Sometimes functions are most conveniently defined by means of differential equations (differential equation). For example, y = sin x is the solution of the differential equation d²y/dx² + y = 0 having y = 0, dy/dx = 1 when x = 0; y = cos x is the solution of the same equation having y = 1, dy/dx = 0 when x = 0.

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