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 *A* = π *r* ^{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.

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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