# function

**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

). In the equationfofx

*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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