function, in mathematics, a relation f that assigns to each member x of some set X a corresponding member y of some set Y; y is said to be a function of x, usually denoted f(x) (read "f of x "). In the equation y=f(x), x is called the independent variable and y the dependent variable. In practice, X and Y will most often be sets of numbers, vectors, points of some geometric object, or the like. For example, X might be a solid body and f(x) the temperature at the point x in X; in this case, Y will be a set of numbers. The formula Ar 2 expresses the area of a circle as a function of its radius. A function f is often described in terms of its graph, which consists of all points (x,y) in the plane such that y=f(x). Although a function f assigns a unique y to each x, several x 's may yield the same y ; e.g., if y=f(x)=x 2 (x is a number), then f(2)=f(-2). If this never occurs, then f is called a one-to-one, or injective, function.

Variable quantity that mathematically describes the wave characteristics of a particle. It is related to the likelihood of the particle being at a given point in space at a given time, and may be thought of as an expression for the amplitude of the particle wave, though this is strictly not physically meaningful. The square of the wave function is the significant quantity, as it gives the probability for finding the particle at a given point in space and time. Seealso wave-particle duality.

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Sentencelike expression that may be thought of as obtained from a sentence by substituting variables for constants occurring in the sentence. For example, “x was a parent of y” may be thought of as obtained from “Adam was a parent of Abel.” A propositional function therefore has no truth-value, becoming true or false only when its free variables are replaced by constants of appropriate syntactic categories (e.g., “Abraham was a parent of Isaac”).

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Equation that expresses the relationship between the quantities of productive factors (such as labour and capital) used and the amount of product obtained. It states the amount of product that can be obtained from every combination of factors, assuming that the most efficient available methods of production are used. The production function can thus measure the marginal productivity of a particular factor of production and determine the cheapest combination of productive factors that can be used to produce a given output.

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In mathematics, one of a set of functions related to the hyperbola in the same way the trigonometric functions relate to the circle. They are the hyperbolic sine, cosine, tangent, secant, cotangent, and cosecant (written “sinh,” “cosh,” etc.). The hyperbolic equivalent of the fundamental trigonometric identity is cosh2math.z − sinh2math.z = 1. The hyperbolic sine and cosine, particularly useful for finding special types of integrals, can be defined in terms of exponential functions: sinhmath.x = (math.emath.xmath.emath.x) ÷ 2 and coshmath.x = (math.emath.x + math.emath.x) ÷ 2

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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 math.A = πmath.r2, for example, assigns to each positive real number math.r the area math.A of a circle with a radius of that length. The symbols math.f(math.x) and math.g(math.x) are typically used for functions of the independent variable math.x. A multivariable function such as math.w = math.f(math.x, math.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 math.f(math.x + math.k) = math.f(math.x) for any value of math.x, math.f is a periodic function with a period of length math.k (a constant). The trigonometric functions are periodic. Seealso density function; exponential function; hyperbolic function; inverse function; transcendental function.

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In mathematics, a function in which a constant base is raised to a variable power. Exponential functions are used to model changes in population size, in the spread of diseases, and in the growth of investments. They can also accurately predict types of decline typified by radioactive decay (see half-life). The essence of exponential growth, and a characteristic of all exponential growth functions, is that they double in size over regular intervals. The most important exponential function is math.emath.x, the inverse of the natural logarithmic function (see logarithm).

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