Chapter 2.1 Overview This chapter deals with linking pair of elements from two sets and then introduce relations between the two elements in the pair. Practically in every day of our lives, we pair the members of two sets of numbers. For example, each hour of the day is paired with the local temperature reading by T.V. Station's weatherman, a teacher often pairs each set of score with the number of students receiving that score to see more clearly how well the class has understood the lesson. Finally, we shall learn about special relations called functions. 2.1.1 Cartesian products of sets Definition : Given two non-empty sets A and B, the set of all ordered pairs (x, y), where x ∈ A and y ∈ B is called Cartesian product of A and B; symbolically, we write A × B = {(x, y) | x ∈ A and y ∈ B} If A = {1, 2, 3} and B = {4, 5}, then A × B = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)} and B × A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} (i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal, i.e. (x, y) = (u, v) if and only if x = u, y = v. (ii) If n(A) = p and n (B) = q, then n (A × B) = p × q. (i) A × A ×A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet. 2.1.2 Relations A Relation R from a non-empty set A to a non empty set B is a subset of the Cartesian product set A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The set of all first elements in a relation R, is called the domain of the relation R, and the set of all second elements called images, is called the range of R. 1For example, the set R = {(1, 2), (– 2, 3), ( , 3)} is a relation; the domain of2 1R = {1, – 2, } and the range of R = {2, 3}.2 (i) A relation may be represented either by the Roster form or by the set builder form, or by an arrow diagram which is a visual representation of a relation. (ii) If n (A) = p, n (B) = q; then the n (A × B) = pq and the total number of possible relations from the set A to set B = 2pq. 2.1.3 Functions A relation f from a set A to a set B is said to be function if every element of set A has one and only one image in set B. In other words, a function f is a relation such that no two pairs in the relation has the same first element. The notation f : X →Y means that f is a function from X to Y. X is called the domain of f and Y is called the co-domain of f. Given an element x ∈ X, there is a unique element y in Y that is related to x.The unique element y to which f relates x is denoted by f (x) and is called f of x, or the value of f at x, or the image of x under f. The set of all values of f (x) taken together is called the range of f or image of X under f. Symbolically. range of f = { y ∈ Y | y = f (x), for some x in X} Definition : A function which has either R or one of its subsets as its range, is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function. 2.1.4 Some specific types of functions (i) Identity function: The function f : R → R defined by y = f (x) = x for each x ∈ R is called the identity function. Domain of f = R Range of f = R (ii) Constant function: The function f : R → R defined by y = f (x) = C, x ∈ R, where C is a constant ∈ R, is a constant function. Domain of f = R Range of f = {C} (iii) Polynomial function: A real valued function f : R → R defined by y = f (x) = a0 + ax + ...+ a xn, where n ∈ N, and a, a, a...a ∈ R, for each x ∈ R, is called1n 01 2n Polynomial functions. ()fx(iv) Rational function: These are the real functions of the type , where()gx f (x) and g (x) are polynomial functions ofx defined in a domain, where g(x) ≠ 0. For RELATIONS AND FUNCTIONS 21 x+1 example f: R – {– 2} → R defined by f(x) = , ∀ x∈ R – {– 2 }is ax+2 rational function. (v) The Modulus function: The real function f: R → R defined by f(x) = x= , ≥ 0⎧xx ⎨, < 0−xx⎩ ∀x∈ R is called the modulus function. Domain of f= R Range of f= R+ ∪ {0} (vi) Signum function: The real function f: R → R defined by ⎧|| ⎧ 1, if x>0 ⎪ x , x≠ 0 ⎪() == 0, if x=0fx ⎨ x ⎨ ⎪⎪⎩ 0, x= 0 ⎩−1, if x<0 is called the signum function. Domain of f= R, Range of f= {1, 0, – 1} (vii) Greatest integer function: The real function f: R → R defined by f(x) = [x], x∈R assumes the value of the greatest integer less than or equal to x, is called the greatest integer function. Thus f(x) =[x] =– 1 for – 1 ≤ x < 0 f(x) =[x] =0 for 0 ≤ x < 1 [x] =1 for 1 ≤ x< 2 [x] =2 for 2 ≤ x< 3 and so on 2.1.5 Algebra of real functions (i) Addition of two real functions Let f: X → R and g: X → R be 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. (ii) Subtraction of a real function from another Let f: X → R and g: X → R be 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. (iii) Multiplication by a Scalar Let f: X → R be a real function and α be any scalar belonging to R. Then the product αfis function from X to R defined by (α f) (x) = α f(x), x∈ X. (iv) Multiplication of two real functions Let f : X → R and g : x → R be any two real functions, where X ⊆ R. Then product of these two functions i.e. f g : X → R is defined by ( f g ) (x) = f (x) g (x) ∀x ∈ X. (v) Quotient of two real function Let f and g be two real functions defined from X → R. The quotient of f by g f denoted by is a function defined from X → R as gff ()⎛⎞() = x , provided g (x) ≠ 0, x ∈ X.x⎜⎟g ()⎝⎠ gx 2.2 Solved Examples Short Answer Type Example 1 Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine (i) A × B (ii) B×A (iii) IsA × B = B ×A ? (iv) Is n (A × B) = n (B × A) ? Solution Since A = {1, 2, 3, 4} and B = {5, 7, 9}. Therefore, (i) A × B = {(1, 5), (1, 7), (1, 9), (2, 5), (2, 7), (2, 9), (3, 5), (3, 7), (3, 9), (4, 5), (4, 7), (4, 9)} (ii) B × A = {(5, 1), (5, 2), (5, 3), (5, 4), (7, 1), (7, 2), (7, 3), (7, 4), (9, 1), (9, 2), (9, 3), (9, 4)} (iii) No, A × B ≠ B × A. Since A × B and B × A do not have exactly the same ordered pairs. (iv) n (A × B) = n (A) × n (B) = 4 × 3 = 12 RELATIONS AND FUNCTIONS 23 n (B × A) = n (B) × n (A) = 4 × 3 = 12 Hence n (A × B) = n (B × A) Example 2 Find x and y if: (i) (4x + 3, y) = (3x + 5, – 2) (ii) (x – y, x + y) = (6, 10) Solution (i) Since (4x + 3, y) = (3x + 5, – 2), so 4x + 3 = 3x + 5 or x =2 and y =–2 (ii) x – y = 6 x + y = 10 ∴ 2x =16 or x =8 8 – y =6 ∴ y =2 Example 3 IfA = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}, a ∈ A, b ∈ B, find the set of ordered pairs such that 'a' is factor of 'b' and a < b. Solution Since A ={2, 4, 6, 9} B = {4, 6, 18, 27, 54}, we have to find a set of ordered pairs (a, b) such that a is factor of b and a < b. Since 2 is a factor of 4 and 2 < 4. So (2, 4) is one such ordered pair. Similarly, (2, 6), (2, 18), (2, 54) are other such ordered pairs. Thus the required set of ordered pairs is {(2, 4), (2, 6), (2, 18), (2, 54), (6, 18), (6, 54,), (9, 18), (9, 27), (9, 54)}. Example 4 Find the domain and range of the relation R given by 6R = {(x, y) : y = x + ; where x, y ∈ N and x < 6}.x Solution When x = 1, y = 7 ∈ N, so (1, 7) ∈ R. Again for, 6 x = 2 . y = 2 + = 2 + 3 = 5 ∈ N, so (2, 5) ∈ R. Again for 2 6 x = 3, y= 3 + = 3 + 2 = 5 ∈ N, (3, 5) ∈ R. Similarly for x= 436 6 y = 4 + 4 ∉ N and for x = 5 , y = 5 + 5 ∉ N Thus R = {(1, 7), (2, 5), (3, 5)}, where Domain of R = {1, 2, 3} Range of R = {7, 5} Example 5 Is the following relation a function? Justify your answer 1 (i) R1 = {(2, 3), ( , 0), (2, 7), (– 4, 6)}(ii) R2 = {(x, |x|) | x is a real number} 2 Solution Since (2, 3) and (2, 7) ∈ R1 ⇒ R (2) = 3 and R (2) = 71 1So R1 (2) does not have a unique image. Thus R1 is not a function. (iii) R2 = {(x, |x|) / x ∈R} For every x∈ R there will be unique image as |x| ∈ R. Therefore R2 is a function. Example 6 Find the domain for which the functions f (x) = 2x2 – 1 and g (x) = 1 – 3x are equal. Solution For f (x) = g (x) ⇒ 2x2 – 1 = 1 – 3x ⇒ 2x2 + 3x – 2 = 0 ⇒ 2x2 + 4x– x – 2 = 0 ⇒ 2x (x + 2) – 1 (x + 2) = 0 ⇒ (2x – 1) (x + 2) = 0 ⎧1 ⎫Thus domain for which the function f (x) = g(x) is ⎨ ,–2 ⎬ . ⎩ 2 ⎭ Example 7 Find the domain of each of the following functions. xfx(i) () = 2 (ii) f (x) = [x] + x x + 3x+2 RELATIONS AND FUNCTIONS 25 Solution ()gx(i) fis a rational function of the form , where g(x) = xand h(x) = x2 + 3x+ 2.()hxNow h(x) ≠ 0 ⇒ x2 + 3x+ 2 ≠ 0 ⇒ (x+ 1) (x+ 2) ≠ 0 and hence domain of the given function is R – {– 1, – 2}. (ii) f(x) = [x] + x,i.e., f(x) = h(x) + g(x) where h(x) = [x] and g(x) = x The domain of h= R and the domain of g= R. Therefore Domain of f= R Example 8 Find the range of the following functions given by x−4 (i) (ii) 16 – x2 x−4 Solution ⎧ x−4 =1, x>4⎪x−4 ⎪x−4 (i) f(x) = = ⎨x−4(x 4) −−⎪ =−1, x<4⎪ x−4⎩ x−4 Thus the range of = {1, –1}.x−4 (ii) The domain of f, where f(x) = 16−x2 is given by [– 4, 4]. For the range, let y= 16 −x2 then y2 = 16 – x2 or x2 = 16 – y2 Since x∈ [– 4, 4] Thus range of f= [0, 4] Example 9 Redefine the function which is given by f(x) = x 11+x−+ , – 2 ≤ x≤ 2 Solution f(x) = x11+x−+, – 2 ≤ x≤ 2 ⎧– x11 x, –2 x–1 +−− ≤< ⎪ +++ ≤< = ⎨– x1 x1, –1 x1 ⎪⎩ x2x−++ 11 x,1 ≤≤ –2,x –2 x–1 ⎧≤<⎪= ⎨ 2, –1 x≤<1 ⎪⎩ 2,1 x 2x≤≤ 1Example 10 Find the domain of the function fgiven by f(x) = []–[] x2 x–6 1Solution Given that f(x) = , fis defined if [x]2 – [x] – 6 > 0.[]–[] x2 x–6 or ([x]–3) ([x] + 2) > 0, ⇒ [x] <–2 or [x] > 3 ⇒ x <–2 or x≥ 4 Hence Domain = ( – ∞, – 2) ∪ [4, ∞). Objective Type Questions Choose the correct answer out of the four given possible answers (M.C.Q.) 1Example 11 The domain of the function fdefined by f(x) = is (A) R (B) R+ –(C) R(D) None of these 1 Solution The correct answer is (D). Given that f(x) = x−x ⎧xx– =0if x≥0where x– =x⎨2x if x<0⎩ RELATIONS AND FUNCTIONS 27 1Thus is not defined for any x ∈ R. x−x Hence f is not defined for any x ∈ R, i.e. Domain of f is none of the given options. 11Example 12 If f (x) = x3 −3 , then f (x) + f ( ) is equal toxx 1(A) 2x3 (B) 2 3 (C)0 (D) 1 x Solution The correct choice is C. 1 Since f (x) = x3 – 3x ⎛⎞1 111 3f = −= – x⎜⎟33 xx 1 x⎝⎠3x ⎛⎞1 311 3Hence, f (x) + f =x −+ – x = 0⎜⎟33⎝⎠x xx Example 13 Let A and B be any two sets such that n(B) = p, n(A) = q then the total number of functions f : A → B is equal to __________. Solution Any element of set A, say xi can be connected with the element of set B in p ways. Hence, there are exactly pq functions. Example 14 Let f and g be two functions given by f = {(2, 4), (5, 6), (8, – 1), (10, – 3)} g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, – 5)} then. Domain of f + g is __________ Solution Since Domain of f = Df = {2, 5, 8, 10} and Domain of g = Dg = {2, 7, 8, 10, 11}, therefore the domain of f + g = {x | x ∈ D f ∩ D g} = {2, 8, 10} Short Answer Type 1. Let A = {–1, 2, 3} and B = {1, 3}. Determine (i) A × B (ii) B×A (iii) B × B (iv) A ×A 2. If P = {x: x< 3, x∈ N}, Q = {x: x≤ 2, x∈ W}. Find (P ∪ Q) × (P ∩ Q), where W is the set of whole numbers. 3. If A = {x: x∈ W, x< 2} B = {x: x∈ N, 1 < x< 5} C = {3, 5} find (i) A × (B ∩ C) (ii) A × (B ∪ C) 4. In each of the following cases, find aand b. ⎛a ⎞ a(i) (2a+ b, a– b) = (8, 3) (ii) ⎜,–2 b⎟= (0, 6 + b)⎝4 ⎠5. Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x∈ A, y∈ A}. Find the ordered pairs which satisfy the conditions given below: (i) x+ y = 5 (ii) x+ y< 5 (iii) x+ y> 8 6. Given R = {(x, y) : x, y∈ W, x2 + y2 = 25}. Find the domain and Range of R. 7. If R1 = {(x, y) | y= 2x+ 7, where x∈ R and – 5 ≤ x≤ 5} is a relation. Then find the domain and Range of R1. 8. If R2 = {(x, y) | xand y are integers and x2 + y2 = 64} is a relation. Then find R2. 9. If R3 = {(x, x) | xis a real number} is a relation. Then find domain and range of R3. 10. Is the given relation a function? Give reasons for your answer. (i) h = {(4, 6), (3, 9), (– 11, 6), (3, 11)} (ii) f= {(x, x) | xis a real number} ⎧ 1 ⎫⎛⎞(iii) g= ⎨n,|n is a positive integer⎜⎟ ⎬⎝⎠⎩ n ⎭ (iv) s= {(n, n2) | nis a positive integer} (v) t= {(x, 3) | x is a real number. 11. If fand gare real functions defined by f(x) = x2 + 7 and g(x) = 3x+ 5, find each of the following 1⎛⎞(a) f(3) + g(– 5) (b) f⎜⎟× g(14)2⎝⎠(c) f(– 2) + g(– 1) (d) f(t) – f(– 2) ft() −f(5) (e) ,if t≠5 t−5 RELATIONS AND FUNCTIONS 29 12. Let fand gbe real functions defined by f(x) = 2x+ 1 and g(x) = 4x– 7. (a) For what real numbers x, f(x) = g(x)? (b) For what real numbers x, f(x) < g(x)? 13. If fand gare two real valued functions defined as f(x) = 2x+ 1, g(x) = x2 + 1, then find. f(i) f+ g (ii) f– g (iii) fg (iv) g 14. Express the following functions as set of ordered pairs and determine their range. f: X → R, f(x) = x3 + 1, where X = {–1, 0, 3, 9, 7} 15. Find the values of xfor which the functions f(x) = 3x2 – 1 and g(x) = 3 + xare equal Long AnswerType 16. Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? Justify. If this is described by the relation, g(x) = αx+ β, then what values should be assigned to α and β? 17. Find the domain of each of the following functions given by 1 1fx(i) () = (ii) ()=fx 1cos x +− x x x3 x 3−+(iii) f(x) = xx (iv) f(x) = 2x−1 3x(v) f(x) = 2x−8 18. Find the range of the following functions given by 3 (i) f(x) = 2 (ii) f(x) = 1 – x−2 2– x (iii) f(x) = x−3 (iv) f(x) = 1 + 3 cos2x (Hint : – 1 ≤ cos 2x≤ 1 ⇒ – 3 ≤ 3 cos 2x≤ 3 ⇒ –2 ≤ 1 + 3cos 2x≤ 4) 19. Redefine the function f(x) = x−2 + 2 +x, – 3 ≤ x≤ 3 x −120. If f (x) = , then show thatx +1 1 11⎛⎞ ⎛⎞ −(i) f = – f (x) (ii) f −=⎜⎟ ⎜⎟⎝⎠x x ()⎝⎠ fx 21. Let f (x) = x and g (x) = x be two functions defined in the domain R+ ∪ {0}. Find (i) (f + g) (x) (ii) (f – g) (x) f⎛⎞(iii) (fg) (x) ⎜⎟()(iv) x g⎝⎠ 1 22. Find the domain and Range of the function f (x) = ax −b23. If f (x)= y = , then prove that f (y) = x. cx−a Objective Type Questions Choose the correct answers in Exercises from 24 to 35 (M.C.Q.) 24. Let n (A) = m, and n (B) = n. Then the total number of non-empty relations that can be defined from A to B is (A) mn (B) nm – 1 2mn(C) mn – 1 (D) – 1 25. If [x]2 – 5 [x] + 6 = 0, where [ . ] denote the greatest integer function, then (A) x ∈ [3, 4] (B) x ∈ (2, 3] (C) x ∈ [2, 3] (D) x ∈ [2, 4) 126. Range of f (x) = is12cos x−⎡⎤ 1 ⎡−1 ⎤(A) ,1 (B) 1,⎢⎥⎢ ⎥3 3⎦⎣⎦ ⎣ ⎡ ⎞ ⎡−⎤(C) (– ∞, –1] ∪ ⎢1,∞⎟ (D) ⎢ 1 ,1⎥⎣3 ⎠ ⎣3 ⎦ RELATIONS AND FUNCTIONS 31 27. Let f (x) = 1+ x2 , then (A) f (xy) = f (x) . f (y) (B) f (xy) ≥ f (x) . f (y) (C) f (xy) ≤ f (x) . f (y) (D) None of these 22 2222[Hint : find f (xy) = 1+ xy , f (x) . f (y) = 1+ xy + x + y ] 28. Domain of a2 −x2 (a > 0) is (A) (– a, a) (B) [– a, a] (C) [0, a] (D) (– a, 0] 29. If f(x) = ax + b, where a and b are integers, f (–1) = – 5 and f (3) = 3, then a and b are equal to (A) a = – 3, b = –1 (B) a = 2, b = – 3 (C) a = 0, b = 2 (D) a = 2, b = 3 1 30. The domain of the function f defined by f (x) = 4 − x + 2 is equal tox −1 (A) (– ∞, – 1) ∪ (1, 4] (B) (– ∞, – 1] ∪ (1, 4] (C) (– ∞, – 1) ∪ [1, 4] (D) (– ∞, – 1) ∪ [1, 4) 4− x31. The domain and range of the real function f defined by f (x) = is given byx −4 (A) Domain = R, Range = {–1, 1} (B) Domain = R – {1}, Range = R (C) Domain = R – {4}, Range = {– 1} (D) Domain = R – {– 4}, Range = {–1, 1} 32. The domain and range of real function f defined by f (x) = x −1 is given by (A) Domain = (1, ∞), Range = (0, ∞) (B) Domain = [1, ∞), Range = (0, ∞) (C) Domain = [1, ∞), Range = [0, ∞) (D) Domain = [1, ∞), Range = [0, ∞) x2 +2 x+1 33. The domain of the function f given by f (x) = 2x – x–6 (A) R – {3, – 2} (B) R – {–3, 2} (C) R – [3, – 2] (D) R – (3, – 2) 34. The domain and range of the function f given by f (x) = 2 – x −5 is (A) Domain = R+, Range = ( – ∞, 1] (B) Domain = R, Range = ( – ∞, 2] (C) Domain = R, Range = (– ∞, 2) (D) Domain = R+, Range = (– ∞, 2] 35. The domain for which the functions defined by f (x) = 3x2 – 1 and g (x) = 3 + x are equal is ⎧ 4 ⎫⎡ 4 ⎤(A) ⎨−1, ⎬ (B) ⎢−1, ⎥⎩ 3 ⎭⎣ 3 ⎦ ⎛ 4 ⎞⎡ 4 ⎞(C) ⎜−1, ⎟ (D) −1, ⎟⎝ 3 ⎠ ⎢⎣ 3 ⎠ Fill in the blanks : 36. Let f and g be two real functions given by f = {(0, 1), (2, 0), (3, – 4), (4, 2), (5, 1)} g = {(1, 0), (2, 2), (3, – 1), (4, 4), (5, 3)} then the domain of f . g is given by _________. 37. Let f = {(2, 4), (5, 6), (8, – 1), (10, – 3)} g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, 5)} be two real functions. Then Match the following : ⎧⎛ 4 ⎞⎛ −1⎞⎛ −3 ⎞⎫2, ,8, ,10, (a) f – g (i) ⎨⎜⎟⎜ ⎟⎜ ⎟⎬ ⎩⎝ 5 ⎠⎝ 4 ⎠⎝ 13 ⎠⎭ (b) f + g (ii) 2,20 , 8, − 4 , 10, − 39{()( )( )} (c) f . g (iii) 2,−1, 8, − 5 , 10, −16 {()( )( )} RELATIONS AND FUNCTIONS 33 f (d) (iv) {(2, 9), (8, 3), (10, 10)}g State True or False for the following statements given in Exercises 38 to 42 : 38. The ordered pair (5, 2) belongs to the relation R = {(x, y) : y = x – 5, x, y ∈ Z} 39. If P = {1, 2}, then P × P × P = {(1, 1, 1), (2, 2, 2), (1, 2, 2), (2, 1, 1)} 40. If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, then (A×B) ∪ (A × C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}. ⎛ 1 ⎞−1441. If (x – 2, y + 5) = 2, are two equal ordered pairs, then x = 4, y =⎜− ⎟ 3⎝ 3 ⎠42. If A × B = {(a, x), (a, y), (b, x), (b, y)}, then A = {a, b}, B = {x, y}