Chapter 2.1 Functions PDF

60 E3 If A and B are two non-empty sets, then a rule f which associated to each x  A, a unique number y  B, is called a function from A to B and we write, f : A B. 2.1.1 Some Important Definitions. (1) Real numbers : Real numbers are those which are either rational or irrational. The set of real n...

60 E3 If A and B are two non-empty sets, then a rule f which associated to each x  A, a unique number y  B, is called a function from A to B and we write, f : A B. 2.1.1 Some Important Definitions. (1) Real numbers : Real numbers are those which are either rational or irrational. The set of real numbers is denoted by R. ID (i) Rational numbers : All numbers of the form p / q where p and q are integers and q  0, are called rational numbers and their set is denoted by Q. e.g. 4 4  are rational 1 U numbers. 5 2  ,  , 4  as 3 2  (ii) Irrational numbers : Those are numbers which can not be expressed in form of p / q are called irrational numbers and their set is denoted by Q c (i.e., complementary set of Q) e.g. D YG 2, 1  3 ,  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,……} Number Chart: Real Number (R) U Rational Numbers (Q) Integers (I or Z) Non-Integers ST Negative Integers (I –) Note : Zero Integers (I0) Irrational Numbers (QC) Positive Integers (I+) or Natural number (N) Whole Numbers (W)  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,…..}  Positive real numbers: R   (0, )  Negative real numbers: R   (, 0)  R 0 : all real numbers except 0 (Zero)  Imaginary numbers: C  {i,  ,....}  Even numbers: E  {0, 2, 4 , 6,......}  Odd numbers: 0  {1, 3, 5, 7,......}  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 60  0 (zero) is a rational number  In irrational numbers, digits are not repeated after decimal   and e are called special irrational quantities   is neither a rational number nor an irrational number E3 (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 ( A  r 2 ) depends upon its radius ID (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. U (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: (ii) D YG (i) Independent variable : A variable which can take any arbitrary value, is called independent variable. Dependent variable : A variable whose value depends upon the independent variable is called dependent variable. e.g. y  x 2 , if x = 2 then y = 4  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 U value throughout a set of mathematical operation. These are generally denoted by a, b, c etc. There are two types of constant. ST (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. 2, 3,  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. y  mx  c 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  x if x  0  | x |   0 if x  0. We also define |x| as follows, |x|= maximum {x, – x} or |x|= x 2  x if x  0  The properties of absolute value are (ii) The inequality | x |  a means x  a 60 (i) The inequality | x |  a means  a  x  a or x  a (iii) x | x|  ,y  0 y | y| E3 (v) (iv) | xy | | x || y | | x  y | | x |  | y | and | x  y | | x |  | y | (6) Greatest integer: Let x  R. Then [x] denotes the greatest integer less than or equal to x; e.g. [1.34]=1, [– 4.57]= – 5, [0.69] = 0 etc. ID (7) Fractional part : We know that x  [x ]. 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, {x}  x  [x ] U e.g., if x = 4.92 then [x] = 4 and {x}= 0.92. D YG Note :  Fractional part of any number is always non-negative and less than one. 2.1.2 Intervals. If a variable x assumes any real value between two given numbers, say a and b (a