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SETS, RELATIONS & FUNCTIONS, , RELATION & FUNCTIONI, 1. INTRODUCTION, , 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, –1, of R, denoted by R , is a relation from B to A and is, defined by, –1, , R = {(b, a) : (a, b) R}, , In this chapter, we will learn how to create a relation between, , –1, , (a, b) R (b, a) R, , two sets by linking pairs of objects from two sets. We will, , Clearly,, , learn how a relation qualifies for being a function. Finally,, , Also, Dom (R) = Range (R ) and Range (R) = Dom (R )., , we will see kinds of function, some standard functions etc., , –1, , –1, , 3. FUNCTIONS, , 2. RELATIONS, 3.1 Definition, , 2.1 Cartesian product of sets, , 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., , Definition : Given two non-empty sets P & Q. The cartesian, product P × Q is the set of all ordered pairs of elements from, , Notations, , P & Q i.e., P × Q = {(p, q); p P; q Q}, 2.2 Relations, 2.2.1 Definition : Let A & B be two non-empty sets. Then, any subset ‘R’ of A × B is a relation from A to B., If (a, b) R, then we 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, called domain & the set of second elements in R is, called range of R. symbolically., Domain of R = { x : (x, y) R}, Range of R = { y : (x, y) R}, The set B is called co-domain of relation R., Note that range co-domain., , 3.2 Domain, Co-domain and Range of a function, , 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, pq, , number of relations is 2 ., , Domain : When we define y = f (x) with a formula and the domain, is not stated explicitly, the domain is assumed to be the largest set, of x–values for which the formula gives real y–values., The domain of y = f (x) is the set of all real x for which f (x) is, defined (real).
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SETS, RELATIONS & FUNCTIONS, , Algo Check : Rules for finding Domain :, (i), , Expression under even root (i.e. square root, fourth root etc.), should be non–negative., , Two functions f & g are said to be equal iff, , (ii), , Denominator z 0., , (iii), , logax is defined when x > 0, a > 0 and a z 1., , (iv), , If domain of y = f (x) and y = g(x) are D1 and D2 respectively,, then the domain of f (x) ± g(x) or f (x) . g(x) is D1 D2. While, domain of, , f x, is D1 D 2 – {x: g(x) = 0}., g x, , 1., , Domain of f = Domain of g, , 2., , Co-domain of f = Co-domain of g, , 3., , f(x) = g(x) x Domain., , 3.3 Kinds of Functions, , Range : The set of all f -images of elements of A is known as the, range of f & denoted by f (A)., Range = f (A) = {f (x) : x A};, f (A)B {RangeCo-domain}., Algo Check : Rule for finding range :, First of all find the domain of y = f (x), If domain finite number of points, , (i), , range set of corresponding f (x) values., If domain R or R – {some finite points}, , (ii), , 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)., If domain a finite interval, find the least and greater value, for range using monotonocity., , (iii), , 1., , Question of format :, , §, ¨y, ©, , Q, ; y, Q, , L, ; y, Q, , Q · Q o quadratic, ¸, L ¹ L o Linear, , Range is found out by cross-multiplying & creating, a quadratic in ‘x’ & making D t 0 (as x R), 2., , Questions to find range 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)., , (a) One-to-One functions are also called Injective, functions., (b) Onto functions are also called Surjective, (c) (one-to-one) & (onto) functions are also called, Bijective Functions.
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SETS, RELATIONS & FUNCTIONS, Relations which can not be catagorized as a function, , 3.4.2 Constant Function : The function f : R o R defined, by y = f(x) = c, x R where c is a constant is, called constant 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), , 3.4.3 Modulus Function : The function f : R o R defined by, , Methods to check one-one mapping, , f (x), 1., , Theoretically : If f (x1) = f (x2), , , is called modulus function. It is denoted by y, = f(x) = | x |., , x1 = x2, then f (x) is one-one., , 2., , Graphically : A function is one-one, iff 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,, , x; x t 0, ®, ¯ x; x 0, , find whether function is only increasing or only, decreasing in their domain. If yes, then function is, one-one, i.e. if f ' x t 0, x domain or i.e.,, , if f ' x d 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 o R defined, by y = f(x) = x x R is called identity function., , Its also known as “Absolute value function’., Properties of Modulus Function :, The modulus function has the following properties :, 1., , 2., , 3., 4., , For any real number x, we have, , xy, , x2, , x y, , x y d x y ½°, ¾ triangle inequality, x y t x y °¿, , x
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SETS, RELATIONS & FUNCTIONS, , 3.4.4, , Signum Function : The function f : R o R defined by, , f (x), , 1; x ! 0, °, ® 0; x 0, ° 1; x 0, ¯, , 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, , (4), , [x] + [– x], , is called signum function. It is usually denoted by, y = f(x) = sgn(x)., , 1, if x I, 0, if x I, , Fractional part of x, denoted by {x} is given by x – [x]. So,, , x, , x x, , x 1; 1 d x 2, °, ® x ; 0 d x 1, ° x 1; 1 d x 0, ¯, , 3.4.6 Exponential Function :, f (x) = ax,, , a > 0, a z 1, , Domain : x R, Range : f(x) (0, f), , Sgn(x), , x, ° ; xz0, ®x, ° 0; x 0, ¯, , 3.4.5 Greatest Integer Function : The function f : R o R, defined as the greatest integer less than or equal to x., It is usually denoted as y = f(x) = [x]
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SETS, RELATIONS & FUNCTIONS, (b) Properties of Monotonocity of Logarithm, , 3.4.7 Logarithm Function :, f (x) = logax,, , a > 0, az1, , Domain : x (0, f), Range : y R, , (i), , If a > 1, loga x < logay, , , , 0 < x <y, , (ii), , If 0 < a < 1, loga x < loga y , , x > y >0, , (iii), , If a > 1 then logax < p, , , , 0 < x < ap, , (iv), , If a > 1 then logax > p, , , , x > ap, , (v), , If 0 < a < 1 then logax < p, , , , x > ap, , (vi), , If 0 < a < 1 then logax > p, , , , 0 < x < ap, , If the exponent and the base are on same side of the, unity, then the logarithm is positive., If the exponent and the base are on different sides of, unity, then the logarithm is negative., , 4. ALGEBRA OF REAL FUNCTION, In this section, we shall learn how to add two real functins,, subtract a real function from another, multiply a real function, by a scalar (here by a scalar we mean a real number), multiply, two real functions and divide one real function by another., 4.1 Addition of two real functions, (a) The Principal Properties of Logarithms, Let M & N are arbitrary positive numbers, a > 0, a z 1,, b > 0, b z 1., a = b, , logb a = c, , (ii), , loga (M . N) = loga M + loga N, , (iii), , loga (M/N) = loga M – loga N, N, , (iv), , loga M = N loga M, , (v), , log b a, , a l ogc b, , (f + g) (x) = f (x) + g(x), for all x X., , c, , (i), , (vi), , Let f : X o R and g : X o R by any two real functions, whre, X R. Then, we define (f + g): X o R by, , l og c a, , c > 0, c z 1., l og c b, , bl ogca , a, b, c > 0, c z 1., , 4.2 Subtraction of a real function from another, Let f : X o R be any two any two real functions, whre X R., Then, we define (f – g): X o R by, (f – g) (x) = f (x) – g(x), for all x X., 4.3 Multiplication by a scalar, Let f : X o R be a real valued function and D be a scalar. Here, by scalar, we mean a real number. Then the product D f is a, function from X to R defined by (D f) (x) = D f(x), x X., 4.4 Multiplication of two real functions, , (a) loga a = 1, (b) logb a . logc b . loga c = 1, (c) loga 1 = 0, (d), , e x ln a, , e ln a, , x, , ax, , The product (or multiplication) of two real functions, f : X o R and g : X o R is a function fg : X o R defined by, (fg) (x) = f(x) g(x), for all x X., This is also called pointwise multiplication.
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SETS, RELATIONS & FUNCTIONS, , 4.5 Quotient of two real functions, Let f and g be two real functions defined from X o R where, X R. The quotient of f by g denoted by f /g is a function, , 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 xT, , defined by, , §f ·, ¨ ¸ x, ©g¹, , Then, f (x) is a periodic function where least positive, value of T is called fundamental period., , f x, , g x , provided g(x) z 0, x X., , 4.6 Even and Odd Functions, , f x , x R., , Graphically : If the graph repeats at fixed interval, then function, is said to be periodic and its period is the width of, that interval., Some standard results on periodic functions :, , Even Function : f (–x) = f (x), x Domain, The graph of an even function y = f (x) is symmetric about the, y–axis. i.e., (x, y) lies on the graph (–x, y) lies on the graph., , Functions, , Periods, S; if n is even., , (i) sinn x, cosn x, secn x, cosecn x, , 2S; (if n is odd or fraction), S; n is even or odd., S, , (ii) tann x, cotn x, (iii) |sin x|, |cos x|, |tan x|, |cot x|, |sec x|, |cosec x|, (iv) x – [x], [.] represents, greatest integer function, (v) Algebraic functions, , 1, period does not exist, , x , x2, x3 + 5, ....etc., , e.g.,, , Properties of Periodic Function, (i), , Odd Function : f (– x) = –f (x), x Domain, , The graph of an odd function y = f (x) is symmetric about origin, i.e. if point (x, y) is on the graph of an odd function, then, (–x, –y) will also lie on the graph., , 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., (ii), , If f (x) is periodic with period T, then, k f (cx + d) has period T/|c|,, i.e. Period is only affected by coefficient of x, where k, c, d constant., , (iii), , If f1(x), f2(x) are periodic functions with periods T1, T2, respectively, then we have, h(x) = a f1(x) + b f2(x) has, period as, LCM of {T1, T2}, , (a), , §a c e·, LCM of ¨ , , ¸, ©b d f ¹, , (b), , LCM of rational and rational always exists., , LCM of a, c, e, HCF of b, d, f, , LCM of irrational and irrational sometime exists., But LCM of rational and irrational never exists., e.g., LCM of (2 S, 1, 6 S) is not possible as, 2 S, 6 S irrational and 1 rational.
