JEE Notes: Relations and Functions notes for JEE Mains and Advanced – Part 2

JEE Notes of Relations and Functions which are important for CBSE Board exam, JEE Mains and JEE Advanced.

Functions: Definition

If A and B are two non-empty sets, then a rule f which associated to each  a unique number is called a function from A to B and we write.

 

Functions: Important Definitions.

(1) Real numbers : Real numbers are those which are either rational or irrational. The set of real numbers is denoted by R.

         (i)  Rational numbers : All numbers of the form  where p and q are integers and q ¹ 0, are called rational numbers and their set is denoted by Q.  e.g. , ,  are rational numbers.

         (ii) Irrational numbers : Those are numbers which can not be expressed in form of  are called irrational numbers  and their set is denoted by (i.e., complementary set of Q) e.g.  are  irrational numbers.

         (iii)     Integers : The numbers …….– 3, – 2, – 1, 0, 1, 2, 3, …….. are called integers. The set of integers is denoted by I or Z. Thus, I or Z ={…….,– 3, – 2, –1, 0, 1, 2, 3,……}

Note :      Set of positive integers I+ = {1, 2, 3, …}

Set of negative integers I= {– 1, – 2, – 3, ……}.       

Set of non negative integers ={0, 1, 2, 3, ..}

Set of non positive integers = {0, – 1, – 2, – 3,…..}

                Prime numbers : The natural numbers greater than 1 which is divisible by 1 and itself only, called  prime numbers.       

              In rational numbers the digits are repeated after decimal

0 (zero) is a rational number

In irrational numbers, digits are not repeated after decimal, pi and e are called special irrational quantities

infinity is neither a rational number nor an irrational number

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         (2) Related quantities : When two quantities are such that the change in one is accompanied by the change in other, i.e., if the value of one quantity depends upon the other, then they are called related quantities. e.g.  the area of a circle depends upon its radius (r) as soon as the radius of the circle increases (or decreases), its area also increases (or decreases). In the given example, A and r are related quantities.

         (3) Variable: A variable is a symbol which can assume any value out of a given set of values. The quantities, like height, weight, time, temperature, profit, sales etc, are examples of variables. The variables are usually denoted by x, y, z, u, v, w, t etc. There are two types of variables mainly:

      (i)     Independent variable : A variable which can take any arbitrary value, is called independent variable.

         (ii) Dependent variable : A variable whose value depends upon the independent variable is called dependent variable. e.g.  if x = 2 then y = 4 Þ so value of y depends on x. y is dependent and x is independent variable here.

         (4) Constant : A constant is a symbol which does not change its value, i.e., retains the same value throughout a set of mathematical operation. These are generally denoted by a, b, c etc. There are two types of constant.

         (i)  Absolute constant : A constant which remains the same throughout a set of mathematical operation is known as absolute constant. All numerical numbers are absolute constants, i.e.  etc. are absolute constants.

         (ii) Arbitrary constant : A constant which remains same in a particular operation, but changes with the change of reference, is called arbitrary constant e.g.  represents a line. Here m and c are constants, but they are different for different lines. Therefore, m and c are arbitrary constants.

         (5) Absolute value : The absolute value of a number x, denoted by |x|, is a number that satisfies the conditions

    We also define |x| as follows, |x|= maximum {x, – x} or |x|=

 

(6)    Greatest integer: Let  Then [x] denotes the greatest integer less than or equal to x; e.g. [1.34]=1,  [– 4.57]= – 5, [0.69] = 0 etc.

         (7) Fractional part : We know that  The difference between the number ‘x’ and it’s integral value ‘[x]’ is called the fractional part of x and is symbolically denoted as {x}. Thus,

e.g., if x = 4.92 then [x] = 4 and {x}= 0.92.

Note : Fractional part of any number is always non-negative and less than one.

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 Interval of Functions

If a variable x assumes any real value between two given numbers, say a and b (a<b) as its value, then x is called a continuous variable. The set of real numbers which lie between two specific numbers, is called the interval.

There are four types of interval:

 

(1) Open interval : Let a and b be two real numbers such that a<b, then the set of all real numbers lying strictly between a and b is called an open interval. It is denoted by ]a, b[ or (a, b)

 

(2) Closed interval : Let a and b be two real numbers such that a<b, then the set of all real numbers lying between a and b including a and b is called a closed interval. It is denoted by [ab]
(3) Open-Closed interval : It is denoted by ]a, b] or (a, b] and ]a, b]  or (a, b]

 

 

 

(4) Closed-Open interval : It is denoted by [a, b[ or [a, b) and    [a, b[ or [a, b)

 

 Definition of Function

(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  where  and  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  is said to be a function if each element in the set X has it’s 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  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 .

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

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