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1. DISCRETE STRUCTURES 23MT1001 Topic: Functions: Basic Concepts, Injective and Bijective Functions Department of Mathematics Session - 4

2. AIM OF THE SESSION To familiarize students with the concepts of real world problems on modelling Functions, Injective and Bijective Functions of the real word problems INSTRUCTIONAL OBJECTIVES This Session is designed to: Demonstrate and Describe the modelling functions of the real word problems and injective and bijective functions . .   LEARNING OUTCOMES At the end of this sessions , Students should be able to: Modelling the functions of the real word problems and solve the problems on injective and bijective functions .

3. SESSION INTRODUCTION In this chapter we focus our attention on a special type of functions, that play an important role in mathematics, computer science, and many applications. We also define some functions used in computer science and examine the growth of functions. 5/20/2023 Koneru Lakshmaiah Education Foundation 3

4. SESSION INTRODUCTION Definition: A function f from a set A to a set B assigns each element of A to exactly one element of B and is denoted by . We write f(a) = b to denote the assignment of b of B to an element a of A by the function f. Here A is called domain of f , B is called co domain of f and the set of images of all elements of A is called the range of f denoted by .Clearly .   5/20/2023 Koneru Lakshmaiah Education Foundation 4

5. SESSION INTRODUCTION 5/20/2023 Koneru Lakshmaiah Education Foundation 5

6. 5/20/2023 Koneru Lakshmaiah Education Foundation 6 Problem 1. Suppose that each student in a discrete mathematics class is assigned a letter grade from the set {A, B, C, D, F}. And suppose that the grades are A for Adams, C for Chou, B for Good friend, A for Rodriguez, and F for Stevens. What is the domain, co domain, and range of the function that assigns grades to students?   Solution: Let G be the function that assigns a grade to a student in our discrete mathematics class. Note that G(Adams) = A, for instance. The domain of G is the set {Adams, Chou, Good friend, Rodriguez, Stevens}, and the co domain is the set {A, B, C, D, F}. The range of G is the set {A, B, C, F}, because each grade except D is assigned to some student. Problem 2. Let f be the function that assigns the last two bits of a bit string of length 2 or greater to that string. For example, f (11010) =10. Then, the domain of f is the set of all bit strings of length 2 or greater, and both the co domain and range are the set {00,01,10,11}.

7. 5/20/2023 Koneru Lakshmaiah Education Foundation 7 Types of functions: One-to-one or Injective function : A function f is said to be one-to-one, or injective, if distinct elements of A are mapped into distinct elements of B , i.e., if implies for all .  

8. SESSION INTRODUCTION Onto function or Surjective function : A function f from A to B is called onto or surjective if for every there is an element such that .   5/20/2023 Koneru Lakshmaiah Education Foundation 8 Problem: Suppose that each worker in a group of employees is assigned a job from a set of possible jobs, each to be done by a single worker. In this situation, the function f that assigns a job to each worker is one-to-one. To see this, note that if x and y are two different workers, then f(x) f(y)because the two workers x and y must be assigned different jobs. The function f is onto if for every job there a worker is assigned this job. The function f is not onto when there is at least one job that has no worker assigned it.

9. SESSION DESCRIPTION   Bijective function: A function f is called a bijection if it is both one-to-one and onto. 5/20/2023 Koneru Lakshmaiah Education Foundation 9

10. SESSION DESCRIPTION (Cont..) 5/20/2023 Koneru Lakshmaiah Education Foundation 10 Problem 1: The total cost of airfare on a given route is comprised of the base cost C and the fuel surcharge S in rupees. Both C and S are functions of the mileage m; C(m) = 0.4m + 50 and S(m) = 0.03m. Determine a function for the total cost of a ticket in terms of the mileage and find the airfare for flying 1600 miles. Solution: Total cost   = C(m) + S(m) = 0.4m + 50 + 0.03m T(m) = 0.43 m + 50 To find the airfare for flying 1600 miles, we have to apply 1600 instead of m. T(m) = 0.43 (1600) + 50  =  688 + 50  =  738 So, the total cost of airfare for flying 1600 miles is 738.

11. 5/20/2023 Koneru Lakshmaiah Education Foundation 11 Problem 2: A salesperson whose annual earnings can be represented by function A(x) = 30, 000 + 0.04x, where x is the rupee value of the merchandise he sells. His son is also in sales and his earnings are represented by the function S(x) = 25, 000 + 0.05x. Find (A + S) (x) and determine the total family income if they each sell Rupees 1, 50, 00, 000 worth of merchandise. Solution: (A + S) (x)   = A(x) + S(x) A(x) = 30, 000 + 0.04x    -------(1) S(x) = 25, 000 + 0.05x    -------(2) (1) + (2) = 30, 000 + 0.04x + 25, 000 + 0.05x   A(x) + S(x) = 55000 + 0.09x. Here x = 15000000    A(x) + S(x) = 55000 + 0.09(15000000) = 55000 + 1350000 Total income = 1405000. So, the required income is 1405000.

12. 5/20/2023 Koneru Lakshmaiah Education Foundation 12 Problem 3: The function for exchanging American dollars for Singapore Dollar on a given day is f(x) = 1.23x, where x represents the number of American dollars. On the same day the function for exchanging Singapore Dollar to Indian Rupee is g(y) = 50.50y, where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee. Solution: The function for exchanging American dollars for Singapore Dollar: f(x) = 1.23x S.D = 1.23 (A.D). Here "x" stands for American dollar and f(x) stands for Singapore dollar. A.D = S.D/1.23  ----(1) Exchanging Singapore Dollar to Indian Rupee is g(y) = 50.50y, I.R = 50.50 (S.D), S.D = I.R/50.50 Applying S.D = I.R/50.50 in (1), we get A.D = (I.R/50.50)/1.23, A.D = I.R/62.115 I.R = 62.115 AD. So, the exchange rate of American dollars in terms of Indian rupee is I.R = 62.115 AD.

13. 5/20/2023 Koneru Lakshmaiah Education Foundation 13 Problem 4: The owner of a small restaurant can prepare a particular meal at a cost of Rupees 100. He estimates that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200−x. Express his day revenue, total cost, and profit on this meal as functions of x. Solution: Cost price of the meal = 100, Selling price = x  Number of customers = 200 - x 1 day revenue = No of customers  ⋅ x = (200 - x)  ⋅ x, 1 day revenue = 200 x - x 2 Total cost = Cost of meal  ⋅ No of customers = 100  ⋅ (200 - x) = 20000 - 100x  Profit = Total cost - 1 day revenue (200 - x)  ⋅ x - 100  ⋅ (200 - x) 

14. 5/20/2023 Koneru Lakshmaiah Education Foundation 14 Problem 5 : Let  A  = {1, 2, 3, 4} and  B  =  N  Let  f :  A  →  B  be defined by  f  ( x ) =  x 3  then, ( i ) find the range of  f   (ii) identify the type of function .

15. ACTIVITIES/ CASE STUDIES/ IMPORTANT FACTS RELATED TO THE SESSION Review questions: What is the function? Define one-one function. Define on-to function. Define bijective function. 5/20/2023 Koneru Lakshmaiah Education Foundation 15

16. SUMMARY In this session, it was discussed about functions, types of functions like one-one, onto and bijective function. 5/20/2023 Koneru Lakshmaiah Education Foundation 16

17. SELF-ASSESSMENT QUESTIONS Q.1.If f(x) = 3x + 2 & g(x) = x 2  – 1. Find f(g(-3)) a) 54 b) 45 c) 46 d) 56 Sol: Option C Explanation:  Now, 1 is neither prime nor composite So f(2(1) + 5) = f(7) = 7 2  + 7 = 56 Q.2.  Given f(x) = x 2  + x; if x is a prime number, f(x) = 2x + 5; if x is non- prime Find f(f(1)) a) 26 b) 29 c) 45 d) 35 Sol: Option A Explanation:  f(g(-3)) = f((-3) 2  -1) = f(8) = 3(8) + 2 = 26 5/20/2023 Koneru Lakshmaiah Education Foundation 17

18. SELF-ASSESSMENT QUESTIONS Q.3.  Given f(x) = x3 + 1, g(x) = 2x – 5, h(x) = [f(x)]2 – g(x) Find h(-2) a) 35 b) 45 d) 56 d) 58 Sol: Option B Explanation:  Now, 3 is an odd number So, g (3 3  – 3 2  + 2) = g(27 – 9 + 2) = g(20) = 2(20) + 5 = 45 Q.4. Given g(x) = x3 – x2 + 2 ; if x is an odd number, g(x) = 2x + 5 ; if x is an even number Find g(g(3)) a) 34 b) -23 c) -12 d) 58 Sol: Option D Explanation: (-2) = [f(-2)] 2  – g(-2) = [(-2) 3  + 1] 2  – [2(-2) -5] [-8 + 1] 2  – [-4-5] = [-7] 2  – [-9] = 49 + 9 = 58 5/20/2023 Koneru Lakshmaiah Education Foundation 18

19. TERMINAL QUESTIONS The total cost of airfare on a given route is comprised of the base cost C and the fuel surcharge S in rupees. Both C and S are functions of the mileage m; C(m) = 0.4m + 50 and S(m) = 0.03m. Determine a function for the total cost of a ticket in terms of the mileage and find the airfare for flying 1600 miles. A salesperson whose annual earnings can be represented by the function A(x) = 30, 000 + 0.04x, where x is the rupee value of the merchandise he sells. His son is also in sales and his earnings are represented by the function S(x) = 25, 000 + 0.05x. Find (A + S) (x) and determine the total family income if they each sell Rupees 1, 50, 00, 000 worth of merchandise. The function for exchanging American dollars for Singapore Dollar on a given day is f(x) = 1.23x, where x represents the number of American dollars. On the same day the function for exchanging Singapore Dollar to Indian Rupee is g(y) = 50.50y, where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee. The owner of a small restaurant can prepare a particular meal at a cost of Rupees 100. He estimates that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200−x. Express his day revenue, total cost and profit on this meal as functions of x. Let  A  = {1, 2, 3, 4} and  B  =  N  Let  f :  A  →  B  be defined by  f  ( x ) =  x 3 then, ( i ) find the range of  f. (ii) identify the type of function. 5/20/2023 Koneru Lakshmaiah Education Foundation 19

20. Tutorial Problems Find the domain and the range of the real function, f(x) = 1/ (x + 3). 2. 5/20/2023 Koneru Lakshmaiah Education Foundation 20 3.Determine whether each of the following is a function with domain {1, 2, 3, 4}. If it is not a function explain, give reason. R 1 = {(1, 1), (2, 1), (3, 1), (4,1), (3, 3)} R 2 = {(1, 2), (2, 3), (4,2)} R 3 = {(1, 1), (2, 1), (3, 1), (4,1)} R 4 = {(1, 4), (2, 3), (3, 2), (4,1)} 4.Determine the domain and range of these functions: The function that assigns to each pair of positive integers the maximum of these two integers The function that assigns the last two bits of a bit string of length 2 or greater to that string.

21. Tutorial Problems 5.Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function: a) the function that assigns to each nonnegative integer its last digit b) the function that assigns the next largest integer to a positive integer c) the function that assigns to a bit string the number of one bit in the string d) the function that assigns to a bit string the number of bits in the string. 5/20/2023 Koneru Lakshmaiah Education Foundation 21

22. Home Assignment Problems 1. Find the domain and range of function f defined by f(x) = x 2  + 1 2. Find the domain and range of the function f(x) = (x 2  +1)/ (x 2  – 1). 3. Let f: Z → Z: f(x) = x 2  and g: Z → Z: g(x) = |x| 2  for all x in Z. Show that f = g 4. Show that f: R – {0} → R – {0}: f(x) = 1/x is a bijection. 5. Let  A  = {−1,1} and B   = {0, 2}. If the function f:  A  →  B  defined by  f  ( x ) =  ax   +  b  is an onto function? Find  a  and  b. 6.The distance  S  an object travels under the influence of gravity in time  t  seconds is given by  S ( t ) = 1/2  gt 2  +  at  +  b,  where g  is the acceleration due to gravity and a ,  b  are   constants. Check if the function  S ( t ) is one-one. 7. The function ‘ t ’ which maps temperature in Celsius ( C ) into temperature in Fahrenheit ( F ) is defined by  t ( C) =  F  where  F  = 9/5 (C )  + 32. Find, ( i )  t (0) (ii)  t (28) (iii)  t (-10) (iv) the value of  C  when  t ( C) = 212 (v) the temperature when the Celsius value is equal to the Fahrenheit value. 5/20/2023 Koneru Lakshmaiah Education Foundation 22

23. REFERENCES FOR FURTHER LEARNING OF THE SESSION TextBooks : 1.Kenneth H. Rosen, Discrete mathematics and its applications, McGraw Hill Publication, 2022. 2.Bernard Kolman , Robert Busby, Sharon C. Ross, Discrete Mathematical Structures, Sixth Edition Pearson Publications, 2015 Reference Books : 1.Joe L Mott, Abraham Kandel, Theodore P Baker, Discrete Mathematics for Computer Scientists and Mathematicians, Printice Hall of India, Second Edition, 2008. 2. Tremblay J P and Manohar R, Discrete Mathematical Structures with Applications to Computer Science, Tata McGraw Hill publishers, 1st edition, 2001,India. Sites and Web links: 1. https://www.youtube.com/watch?v=cqSZnON00OQ 2. https://www.youtube.com/watch?v=ToGSk6zIm6s 5/20/2023 Koneru Lakshmaiah Education Foundation 23

24. THANK YOU Team – Discrete Structures 5/20/2023 Koneru Lakshmaiah Education Foundation 24