JEE Main & Advanced Mathematics Functions Definition of Function

Definition of Function

Category : JEE Main & Advanced

(1) Function can be easily defined with the help of the concept of mapping. Let \[X\] and \[Y\] be any two non-empty sets. “A function from \[X\] to \[Y\] is a rule or correspondence that assigns to each element of set \[X,\] one and only one element of set \[Y''\]. Let the correspondence be \['f'\] then mathematically we write \[f:X\to Y\] where \[y=f(x),\,x\in X\] and \[y\in Y.\] We say that \['y'\] is the image of \['x'\] under \[f\] (or \[x\] is the pre image of \[y\]).



Two things should always be kept in mind:



(i) A mapping \[f:X\to Y\] is said to be a function if each element in the set \[X\] has its image in set \[Y\]. It is also possible that there are few elements in set \[Y\] which are not the images of any element in set \[X\].


(ii) Every element in set \[X\] should have one and only one image. That means it is impossible to have more than one image for a specific element in set \[X\]. Functions can not be multi-valued (A mapping that is multi-valued is called a relation from \[X\] and \[Y\]) e.g.




(2) Testing for a function by vertical line test : A relation \[f:A\to B\] is a function or not it can be checked by a graph of the relation. If it is possible to draw a vertical line which cuts the given curve at more than one point then the given relation is not a function and when this vertical line means line parallel to Y-axis cuts the curve at only one point then it is a function. Figure (iii) and (iv) represents a function.


(3) Number of functions : Let \[X\] and \[Y\] be two finite sets having \[m\] and \[n\] elements respectively. Then each element of set \[X\]can be associated to any one of \[n\] elements of set \[Y\]. So, total number of functions from set \[X\] to set \[Y\] is \[{{n}^{m}}\].    


(4) Value of the function : If \[y=f(x)\] is a function then to find its values at some value of \[x,\] say \[x=a,\] we directly substitute \[x=a\] in its given rule \[f(x)\] and it is denoted by \[f(a)\].    


e.g. If \[f(x)={{x}^{2}}+1,\] then \[f(1)={{1}^{2}}+1=2,\] \[f(2)={{2}^{2}}+1=5,\] \[f(0)={{0}^{2}}+1=1\] etc.  

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