Function

DEFINITION1: Let  X and  Y be two sets and  f\subset{X\times{Y}} be a relation. If the following two conditions are provided, then the relation  f is called a “function” with domain  X and codomain  Y and denoted by  f:X\to{Y} or  X\stackrel{f}{\rightarrow}{Y}.

1.  \forall{x}\in{X}, \exists{y}\in{Y}: (x,y)\in{f},

2.  (x,y),(x,y')\in{f}\Rightarrow{y=y'}.

Henceforth, when we write  f:X\to{Y}, we will consider that “ f is a function from  X to  Y”.

As is seen from the definition, a function is a relation mapping each element in the domain to a unique element in the codomain. So, the notation  y=f(x) is generally used instead of the notations  xfy and  (x,y)\in{f} and read “ x maps to  y” or “ x maps to  f(x)”. The notation  f(x) is read “ f of  x”. Each element of the domain is called an “argument” and for each  x in the domain, the corresponding unique element  y in the codomain is called “the function value at  x”, “output  f for an element  x” or “the image  x” under the function  f. The set defined as  \{f(x)\:|\:x\in{X}\}\subset{Y} is called the “image” or the “range” of  f. Sometimes a function is called a “map” or a “mapping”.

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