# 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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