**In this chapter we would be dealing with one of the most fundamental concept in Mathematics, namely the notion of function.**

**Roughly speaking, term function is used to define the dependence of one physical quantity on another**

**e.g. volume ‘ V ‘ of a sphere of radius ‘ r ‘ is given by**

**V = (4/3)πr ^{3}.**

**This dependence of ‘ V ‘ on ‘ r ‘ would be denoted as V = f(r) and we would simply say that ‘ V ‘ is a function of ‘ r ‘. Here ‘ f ‘ is purely a symbol (for that matter any other letter could have been used in place of ‘ f ‘ ) , and it is simply used to represent the dependence of one quantity on the other.**

__Definition of Function:__

__Definition of Function:__**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 → Y where y = f(x) , x ∈ X and y ∈ 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:**

**⋄ A mapping f: X → Y Domain , Co-domain , Range Y is said to be a function if each element in the set X has it’s image in set Y.**

**It is possible that a few elements in the set Y are present which are not the images of any element in set X.**

**⋄ 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’t be multi-valued (A mapping that is multi-valued is called a relation from X to Y)**

**Let us consider some other examples to make the above mentioned concepts clear.**

**cube of any two distinct real numbers are distinct). Hence it would represent a function.**

**(i) Let f: R+ −> R where y ^{2} = x. This can’t be considered a function as each**

**x ∈ R+ would have two images namely ±√x. Thus it would be a relation.****(ii) Let f : [ −2 , 2 ] −> R , where x ^{2} + y^{2} = 4. Here y = ±√(4-x^{2}), that means for every x∈[-2, 2] we would have two values of y (except when x = ± 2). Hence it does not represent a function.**

**(iii) Let f: R −> R where y = x ^{3}. Here for each x ∈ R we would have a unique value of y in the set R**

**Distinction between a relation and a function can be easily made by drawing the graph of y = f(x).**

**These figures show the graph of two arbitrary curves. In fig.(a) any line drawn parallel to y-axis would meet the curve at only one point. That means each element of X would have one and only one image. Thus fig (a) would represent the graph of a function.**

**In fig.(b) certain line (e.g. line L) would meet the curve in more than one points (A, B and C). Thus element xo of X would have three distinct images. Thus this curve will not represent a function.**

**Hence if y = f (x) represents a function, lines drawn parallel to y-axis through different points corresponding to points of set X should meet the curve in one and only one point.**