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)πr3.
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:
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 y2 = 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 x2 + y2 = 4. Here y = ±√(4-x2), 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 = x3. 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.