Relations and Functions Class 11 Notes CBSE Maths Chapter 2 PDF

Revision Notes Class 11 Mathematics Chapter 2 - Relation & Function-I 1. INTRODUCTION: In this chapter, we'll learn how to link pairs of objects from two sets to form a relation between them. We'll see how a relation can be c...

Revision Notes Class 11 Mathematics Chapter 2 - Relation & Function-I 1. INTRODUCTION: In this chapter, we'll learn how to link pairs of objects from two sets to form a relation between them. We'll see how a relation can be classified as a function. Finally, we'll look at several types of functions, as well as some standard functions. 2. RELATIONS: 2.1 Cartesian product of sets Definition: Given two non-empty sets P and Q. The Cartesian product P  Q is the set of all ordered pairs of elements from P and Q that is P×Q = {( p, q ) ; p  P ; q  Q} 2.2 Relation: 2.2.1 Definition: Let A and B be two non-empty sets. Then any subset ‘ R ’ of A×B is a relation from A and B. If ( a, b ) R , then we can write it as a R b which is read as a is related to b ‘by the relation R ’, ‘ b ’ is also called image of ‘ a ’ under R. 2.2.2 Domain and range of a relation: If R is a relation from A to B , then the set of first elements in R is known as domain & the set of second elements in R is called range of R symbolically. Domain of R = { x:( x, y )  R} Class IX Mathematics www.vedantu.com 1  Range of R = { y:( x, y )  R} The set B is considered as co-domain of relation R. Note that, range  co-domain Note : Total number of relations that can be defined from a set A to a set B is the number of possible subsets of A×B. If n ( A ) = p and n ( B ) = q , then n ( A×B ) = pq and total number of relations is 2pq. 2.2.3 Inverse of a Relation: Let A, B be two sets and let R be a relation from a set A to set B. Then the inverse of R denoted as R -1 , is a relation from B to A and is defined by R –1 = {( b, a ):( a, b )  R} Clearly ( a, b )  R  ( b, a )  R –1 Also, Dom ( R ) = Range(R –1 ) and Range ( R ) = Dom(R –1 ) 3. FUNCTIONS: 3.1 Definition: A relation ‘ f ’ from a set A to set B is said to be a function if every element of set A has one and only one image in set B. Class IX Mathematics www.vedantu.com 2  Notations: 3.2 Domain, Co-domain and Range of a function: Domain: The domain is believed to be the biggest set of x - values for which the formula provides real y - values when y = f ( x ) is defined using a formula and the domain is not indicated explicitly. The domain of y = f (x) is the set of all real x for which f (x) is defined (real). Class IX Mathematics www.vedantu.com 3  Rules for finding Domain: 1. Even roots (square root, fourth root, etc.) should have non–negative expressions. 2. Denominator  0 3. log a x is defined when x > 0, a > 0 and a  1 4. If domain y = f(x) and y = g(x) are D1 and D 2 respectively then the domain of f (x)  g(x) or f (x).g(x) is D1  D 2. f (x) While domain of is D1  D 2 − x : g(x) = 0 g(x) Range: The set of all f - images of elements of A is known as the range of f and can be denoted as f (A). Range = f (A) = f (x) : x  A f(A)  B {Range  Co-domain} Rule for finding range: First of all find the domain of y = f (x) i. If domain  finite number of points  range  set of corresponding f (x) values. ii. If domain  R or R − {Some finite points} Put y = f (x) Then express x in terms of y.From this find y for x to be defined. (i.e., find the values of y for which x exists). iii. If domain  a finite interval, find the least and greater value for range using monotonicity. Note: 1. Question of format:  Q L Q  = = = L  y ; y ; y  Q Q Q → Quadratic Class IX Mathematics www.vedantu.com 4  L → Linear Range is found out by cross-multiplying & creating a quadratic in 'x' & making D  0 (as x  R ) 2. Questions to determine the range of values in which the given expression y = f (x) can be converted into x (or some function of x = expression in ‘ y ’. Do this & apply method (ii). 3. Two functions f & g are said to be equal if a. Domain of f = Domain of g b. Co-domain of f = Co-domain of g c. f (x) = g(x) x  Domain 3.3 Kinds of functions: Class IX Mathematics www.vedantu.com 5  Note: Injective functions are called as one-to-one functions. Surjective functions are also known as onto functions. Bijective functions are also known as (one-to-one) and (onto) functions. Relations which cannot be categorized as a function: As not all elements of set A are associated with some elements of set B. (Violation of– point (i) – definition 2.1 ) An element of set A is not associated with a unique element of set B , (violation of point (ii) definition 2.1 ) Methods to check one-one mapping: 1. Theoretically: f (x1 ) = f (x 2 )  x1 = x 2 then f (x) is one-one. Class IX Mathematics www.vedantu.com 6  2. Graphically: A function is one-one, if no line parallel to x − axis meets the graph of function at more than one point. 3. By Calculus: For checking whether f (x) is One-One, find whether function is only increasing or only decreasing in their domain. If yes, then function is one-one, that is if f '(x)  0, x  domain or, if f '(x)  0, x  domain, then function is one-one. 3.4 Some standard real functions & their graphs: 3.4.1 Identity Function: The function f : R → R defined by y = f ( x ) = xx  R is called identity function. 3.4.2 Constant function: The function f : R → R defined by y = f ( x ) = c, x  R Class IX Mathematics www.vedantu.com 7  3.4.3 Modulus function: The function f : R → R defined by  x; x  0 f (x) =  − x; x  0 is called modulus function. It is denoted by y = f (x) = x It is also known as “Absolute value function”. Properties of Modulus Function: The modulus function has the following properties: 1. For any real number x , we have x 2 = x 2. xy = x y Class IX Mathematics www.vedantu.com 8  3. x+y  x + y Triangle inequality 4. x−y  x − y Triangle inequality 3.4.4 Signum Function: The function f : R → R define by  1: x  0  f (x) =  0 : x = 0 is called signum function. −1: x  0  It is usually denoted as y = f (x) = sgn(x) Note: x  ; x0 sgn(x) =  x  0; x = 0  3.4.5 Greatest Integer Function: The function f : R → R defined as the greatest integer less than or equal to x. It is usually denoted as y = f ( x ) =  x . Class IX Mathematics www.vedantu.com 9  Properties of Greatest Integer Function: If n is an integer and x is any real number between n and n + 1 , then the greatest integer function has the following properties: 1.  − n  = −  n  2.  x + n  =  x  + n 3.  − x  =  x  − 1 −1, if x  I 4.  x  +  −x  =   0, if x  I Note: Fractional part of x , denoted by x is given by x –  x  , Hence  x − 1; 1  x  2 x = x −  x  =  x 0  x 1  x + 1 −1  x  0  3.4.6 Exponential Function: f (x) = a x , a  0,a  1 Domain: x  R Range: f (x)  ( 0,  ) Class IX Mathematics www.vedantu.com 10  3.4.7 Logarithm Function: f (x) = log a x , a  0,a  1 Domain: x  ( 0,  ) Range: y  R Class IX Mathematics www.vedantu.com 11  a) The Principal Properties of Logarithms: Let M and N be the arbitrary positive numbers, a > 0, a  1, b > 0, b  1 1) log b a = a  a = bc 2) log a ( M.N ) = log a M + log a N M 3) log a   = log a M − log a N N 4) log a M N = N log a M log c a 5) log b a = ,c  0,c  1 log c b 6) a logc b = blogc a ,a,b,c  0,c  1 Note: log a a = 1 log b a. log c b. log a c = 1 log a 1 = 0 x e xlna = e xlna = a x b) Properties of Monotonicity of Logarithm: 1) If a > 1, log a x < log a y 0x y 2) If 0 < a < 1, log a x < log a y  x > y > 0 3) If a > 1 then log a x < p  0 < x < ap Class IX Mathematics www.vedantu.com 12  4) If a > 1 then log a x > p  x > ap 5) If 0 < a < 1 then log a x < p  x > ap 6) If 0 < a < 1 then log a x > p  0 < x < ap Note: The logarithm is positive if the exponent and base are on the same side of unity. The logarithm is negative if the exponent and base are on opposite sides of unity. 4. ALGEBRA OF REAL FUNCTION: We'll learn how to add two real functions, remove one from another, multiply a real function by a scalar (a scalar is a real integer), multiply two real functions, and divide one real function by another in this part. 4.1 Addition of two real functions: Let f : X → R and g : X → R by any two real functions, where x  R. Then, we define ( f + g ) : X → R by ( f + g )( x ) = f ( x ) + g ( x ) for all x  X. 4.2 Subtraction of a real function from another: Let f : X → R be any two any two real functions, where x  R. Then, we define ( f − g ) : X → R by ( f − g )( x ) = f ( x ) − g ( x ) for all x  X. 4.3 Multiplication by a scalar: Let f : X → R be a real valued function and  be a scalar. Here by scalar, we mean a real number. Then the product f is a function from X to R defined as ( f )( x ) = f ( x ) , x  X. Class IX Mathematics www.vedantu.com 13  4.4 Multiplication of two real functions: The product (or multiplication) of two real functions f : X → R and g : X → R is a function fg : X → R defined as ( fg )( x ) = f ( x ) g ( x ) for all x  X. This is also known as pointwise multiplication. 4.5 Quotient of two real functions: Let f and g be two real functions defined from X → R where X  R. f f  f (x) The quotient of f by g denoted by a is a function defined as g ( x ) = g   g(x) Provided g ( x )  0, x  X. 4.6 Even and Odd Functions Even function: o f ( − x ) = f ( x ) , x  Domain o The graph of an even function y = f ( x ) is symmetric about the y − axis, that is ( x, y ) lies on the graph  ( − x, y ) lies on the graph. Class IX Mathematics www.vedantu.com 14  Odd Function: o f ( x ) = −f ( x ) , x  Domain o The graph of an odd function y = f ( x ) is symmetric about origin that is if point ( x, y ) is on the graph of an odd function, then ( − x, − y ) will also lie on the graph. 5. PERIODIC FUNCTION Definition: A function f (x) is said to be periodic function, if there exists a positive real number T , such that f ( x + T ) = f ( x ) , x  R Then, f (x) is a periodic function where least positive value of T is called fundamental period. Graphically: The function is said to be periodic if the graph repeats at a set interval, and its period is the width of that interval. Class IX Mathematics www.vedantu.com 15  Some standard results on periodic functions: Functions Periods i sin n x, cos n x, sec n x, cosec n x  ; if n is even 2 ;(if n is odd or fraction) ii tan n x, cot n x  ; n is even or odd iii sinx , cosx , tanx ,  cotx , secx , cosecx iv x −  x  , . represents greatest 1 integer function v Algebraic functions for example Period does not exist x , x 2 , x 3 +5,..... etc. Properties of Periodic Function: i. If f ( x ) is periodic with period T , then a) c.f ( x ) is periodic with period T b) f ( x  c ) is periodic with period T c) f ( x )  c is periodic with period T Where c is any constant i. If f ( x ) is periodic with period T , then T kf ( cx + d ) has period c That is Period can be only affected by coefficient of x where k, c, d constant. ii. If f1 ( x ) ,f 2 ( x ) are periodic functions with periods T1 ,T2 respectively, Then we have, h ( x ) = af1 ( x )  bf 2 ( x ) has period as, LCM of T1 ,T2  Class IX Mathematics www.vedantu.com 16  Note:  a c e  LCM of (a,c,e) a. of  , ,  =  b d f  HCF of (b,d,f ) b. LCM of rational and rational always exists. LCM of irrational and irrational sometime exists. But LCM of rational and irrational never exists. For example, LCM of ( 2π, 1, 6π ) is not possible because 2π, 6π irrational and 1 rational. Class IX Mathematics www.vedantu.com 17