2.2 Functions

Function is a special relation. It is also conceived as a “rule”, because function is a relation between elements of two sets, following certain rule. Every element of a set (say A) is related to exactly one element of other set (say B). This relationship is described as mapping of all elements of one set to elements of another set.

In order to emphasize, we need to enumerate the way “function” relation is special :

• Every element of set “A” is related to elements in set “B”.
• An element of set “A” is related to exactly one element of “B”.

It can be deduced from the above characterization of a function that the element of set “B” may be paired with none or one or more elements of set “A”.

In order to illustrate function relation, let us consider an example. Let "A" and "B" be two sets as given here :

$$A=\{-\mathrm{1,0,1,2,3}\}$$

$$B=\{-\mathrm{1,}\mathrm{0,1,2,3,4,5,6,7,8,9,10}\}$$

The two sets are related by the relation :

$$R=\{\left(x,y\right):y=x^2-\mathrm{1,}x\in A,y\in B\}$$

The values of “y” for given values of “x” are :

$$Forx=-\mathrm{1,}y=1-1=0$$

$$Forx=\mathrm{0,}y=0-1=-1$$

$$Forx=\mathrm{1,}y=1-1=0$$

$$Forx=\mathrm{2,}y=4-1=3$$

$$Forx=\mathrm{3,}y=9-1=8$$

The relation between two sets is pictorially shown with arrow diagram. We note that all elements of “A” are mapped. Further, elements in “A” are uniquely mapped i.e. they are paired to exactly one element of set “B”. It is, however, possible that few of the elements in set “A” are related to same element in “B” like "-1" and "1" in set "A" are related to "0" in set “B”. In the nutshell, we see that this relation meets both properties as enumerated for a function relation and hence is a function relation.

Looking in reverse direction, we see that elements in “B” may be paired – with no element (1,2,4,5,6,7,9,10) or with one element (-1,3,8) or with more than one element (0) in "A".

We generally drop word “relation” from “function relation” and call it simply as “function”. The function is denoted by a small letter like “f”. To elaborate the direction of function, we expand the symbol as :

$$f:A\to B$$

This means that function is mapped from “A” to “B”. Now, in order to define the function, we need to understand the concept of “image” and “pre-image” elements. We call first element “x” of set “A” in the ordered pair (x,y) of the function as the “pre-image” of second element “y” of set “B”. The second element “y” of set “B” is called the “image” of the first element “x” of set “A”.

The image is also denoted as “f(x)”. We read “f(x)” as image of “x” under rule “f”. For a particular value of x = a, "f(a)" is a particular instance of image :

$$f\left(a\right)=b$$

Function

A relation “f” is a function, if every element in set “A” has one and only one image in set “B”.

Domain, range and co-domain of function

As all the elements of set “A” are involved, it emerges that that the set of first elements in the ordered pairs i.e. domain set is same as set “A”. We can not say the same for set "B". The set “B” may have other elements than those, which have been mapped with the elements of “A”. The range is simply the set of images of the function. However, as defined earlier, the set “B” is co-domain of the relation and hence that of function in this special case. It is clear that range is a subset of co-domain "B".