Composite Function

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composite function

[kəm′päz·ət ′fəŋk·shən]
A function of one or more independent variables that are themselves functions of one or more other independent variables.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Composite Function


a function of a function. Let us suppose that the variable y is a function of u, that is, y = f(u), and that u is in turn a function of x, that is, u = Φ(x). Then y is a composite function of u and x—that is, y = f[Φ(x)]—defined for all x such that Φ(x) is in the domain of f(u). In this case, y is said to be a composite function of the variable x and the variable u, which is sometimes called the intermediate variable. For example, if y = u2 and u = sin x, then y = sin2x for all values of x. If, however, Composite Function and u = sin x, then Composite Function, which, if we are restricted to real values of the function, is defined only for all x such that sin x ≥ 0—that is, for 2kπx ≥ (2k + 1)π, where k = 0, ± 1, ± 2,….

The derivative of a composite function is equal to the product of the derivative of the function with respect to the intermediate variable and the derivative of the intermediate variable with respect to the independent variable. This rule, called the chain rule, extends to composite functions with two, three, or more intermediate variables. Thus, if y = f(u1, u1 = Φ(u2), …, uk–1 = Φk–1(uk), uk = Φk(x), then

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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As x = x[q([alpha])] that by a rule of differentiation of the composite function, assuming that rank-[partial derivative]q/[partial derivative][[alpha].sup.T] = m and matrix [partial derivative]q/[partial derivative][[alpha].sup.T] is non-singular, we will receive: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].