1. The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 17 Part 2

2. Introduction to… Sets, Functions, and Graphs Introduction to… Sets, Functions, and Graphs © 2007 Herbert I. Gross next

3. © 2007 Herbert I. Gross next Whether it's the word “function” itself, the ominous notation “f(x)” or yet some other reason, the fact is that the mathematical concept of a function causes many students undue difficulty. Function ------------ f (x) The truth of the matter is that we are surrounded by the concept of functions almost everywhere we turn; only the term “function” is never used. next

4. © 2007 Herbert I. Gross next While the concept of a mathematical function may seem a bit “threatening” to you, it is simply a rule that assigns to members in one set a member of another (or possibly, the same) set. In this sense, something as simple as a mathematics teacher assigning grades to students is a function. Namely in this case, the teacher serves as the function that assigns to each student in her class one (and only one) grade in the mathematics course. next

5. © 2007 Herbert I. Gross next So for example, suppose Tom, Bill and Jane are in a special math honors program. Jane receives a grade of A; Tom receives a grade of A - and Bill receives a grade of B +. We might choose to let S denote the set of students whose members are Tom, Bill and Jane; and we might choose G to denote the set of grades, A, A -, and B +. A-B+

6. © 2007 Herbert I. Gross next As shown below, we might then use arrows to indicate the grade that each student received. Tom Bill Jane A- B+ A In essence, the arrows (which represent the teacher) are the function that assigns to each member in S one and only one member in G. next SG

7. © 2007 Herbert I. Gross next To introduce the language of functions, we might define f t o be the rule that assigns to each member of the set S a member of the Set G. In this case, we might amend the Figure below by rewriting it as… Tom Bill Jane A- B+ A Tom Bill Jane A- B+ A f f f next

8. © 2007 Herbert I. Gross next However, to represent this in even more concise terms, the notation that mathematicians often prefer to use is… f(Tom) = A- next f(Bill) = B+ f(Jane) = A Translated into “plain English” the above is read as “f is the rule which assigns to Tom the grade of A-; to Bill, the grade of B+; and to Jane the grade of A”.

9. © 2007 Herbert I. Gross next In the above example, we chose the letter f only because it suggests the word function. In reality, however, it makes no difference what letter we choose to use. For example, we might have chosen the letter g to remind us that we are dealing with grades. Note

10. © 2007 Herbert I. Gross next The ancient Greeks and Hebrews had an interesting way of writing numbers. They used letters of the alphabet in the following way. The first nine letters represented the numbers 1 through 9; the next nine letters represented the multiples of ten, 10 through 90; and the next nine letters represented the multiples of a hundred, 100 through 900. Gematria: An Interesting Historical Example

11. © 2007 Herbert I. Gross next This led to a form of mysticism known as gematria in which they used the numerical sum of the letters in a word to code the word. Since the Greek alphabet had only 24 letters, they invented three additional symbols in order to have the required 27 symbols. ΑΒΓΔΕΖ ΗΘΙΚΛΜ ΝΞΟΠΡΣ ΤΥΦΧΨΩ

12. © 2007 Herbert I. Gross next In essence, gematria was a rule that assigned numerical values to the set of Greek letters of the alphabet. a =1b = 2c = 3d = 4e = 5f = 6g = 7h = 8i = 9 j = 10k = 20l = 30m = 40n = 50o = 60p = 70q = 80r =90 s = 100t = 200u = 300v = 400w = 500x = 600y = 700z = 800 For example, if we applied gematria to the English alphabet we might have…

13. © 2007 Herbert I. Gross next a =1b = 2c = 1d = 4e = 5f = 6g = 7h = 8i = 9 j = 10k = 20l = 30m = 40n = 50o = 60p = 70q = 80r =90 s = 100t = 100u = 300v = 400w = 500x = 600y = 700z = 800 ? = 900 Thus by gematria, the word “cat” would be coded as 204 because c = 3, a = 1 and t = 200; and 1 + 3 + 200 = 204. In the language of functions, we might define g to be the function (rule) that in terms of gematria assigns to the word “cat” the number 204; and write this as g(cat) = 204. a = 1c = 3 t = 200

14. © 2007 Herbert I. Gross next Gematria led to an interesting superstition. The feeling was that a person's strength was proportional to the numerical sum of the letters in the person's name. Historical Anecdote g(JOHN) = 10 + 60 + 8 + 50 = 128 and g(BILL) = 2 + 9 + 30 + 30 = 71. Hence, if there was a battle between John and Bill, John would be a “128 to 71” favorite. For example next

15. © 2007 Herbert I. Gross next Obviously, this was simply folk lore. For example, to become the favorite, Bill would only have to enter the battle under his full name, William; in which case the W by itself would ensure that his “power” would be at least 500. It is said that the Greek poet, Homer, used gematria in the I liad. Namely, in every battle the numerical sum of the Greek letters in the winner's name was always greater than the numerical sum of the letters in the loser's name.

16. © 2007 Herbert I. Gross next While the above discussion may seem interesting, a natural question might be “But what does all this talk about functions have to do with algebra?” Part of the answer lies in the fact that we were essentially dealing with functions in our study of formulas. For example, consider the formula that relates the perimeter of a square to the length of one of its sides. Namely, we start with the length of a side and then multiply it by 4 to find the perimeter of the square.

17. © 2007 Herbert I. Gross next So, for example, if we let P denote the perimeter of the square in inches and s the length of one of its sides in inches, the formula becomes… P = 4s Using our notation for functions we can rewrite the above formula in the form… P = f(s), and f(s) = 4s s s s s 4s next

18. © 2007 Herbert I. Gross next We read f(s) = 4s, as “f is the function (i.e., rule) which assigns to any number s the value 4s” and in terms of our input/output machine, we would write… Input (s) Program f Where f(s) = 4s Output f(s) = P

19. © 2007 Herbert I. Gross next With respect to the diagram above, in dealing with functions we do not talk about the set of inputs. Rather we refer to them as the domain of f (usually abbreviated as dom f). Nor do we talk about the set of outputs. Rather we refer to them as the image (or the range) of f. Input (s) Program f Where f(s) = 4s Output f(s) = P Domain & Image (Range)

20. © 2007 Herbert I. Gross next In the “old days”, we referred to the set of inputs as being the independent variable (because we were free to choose it at will) and to the set of outputs as the dependent variable (because the output depended on the input). This terminology is still in use today. Historical Note

21. Some Special Domains In many mathematical situations, the domain of a function is a “connected interval”. For example, when we say that it costs 42 cents to mail a first class letter, we mean that it costs 42 cents for up to and including 1 ounce. So if we were to let C denote the cost in cents and w to denote the weight of letters up to and including 1 ounce, the rule would be… next © 2007 Herbert I. Gross C = f(w) where f(w) = 42 w, and the domain of f would be the set: {w: 0 ≤ w ≤ 1}. next 42

22. More generally, if a < b, the notation [a,b] stands for the set {x: a ≤ x ≤ b}. which we read as…“the closed interval from a to b”, and we refer to a and b as the end points of the interval. next © 2007 Herbert I. Gross Closed Interval Rather than write {w: 0 ≤ w ≤ 1} mathematicians often use the notation [0,1] and refer to it as a closed interval because it includes the endpoints 0 and1. Some Mathematical Notation

23. Geometric Illustration of a Closed Interval next © 2007 Herbert I. Gross S = {x:5 ≤ x ≤ 7} =[5,7] 5 7 S -3-3 -2-2 -1 0 12 3 4 5 6 7 89 10 1112 [ ] 7 5 next

24. To represent a closed interval geometrically, some authors prefer to use a solid dot rather than a bracket to represent the endpoints. © 2007 Herbert I. Gross Thus, rather than writing… -3-3 -2-2 -1 0 12 3 4 5 6 7 89 10 1112 [ ] 7 5 they would write… -3-3 -2-2 -1 0 12 3 4 5 6 7 89 10 1112 7 5 next Author’s Preference

25. next © 2007 Herbert I. Gross If we don’t want to include the end points of an interval, we write (a,b) and call it the open interval from a to b. (a,b) stands for the set: {x: a < x < b} More Mathematical Notation Open Interval

26. Geometric Illustration of an Open Interval T = {x: 5 < x < 7} =(5,7) T -3-3 -2-2 -1 0 12 3 4 5 6 7 89 10 1112 ( ) 7 5 next © 2007 Herbert I. Gross

27. To represent an open interval geometrically, some authors prefer to use hollow dots rather than parentheses to represent the endpoints. next © 2007 Herbert I. Gross Thus, rather than writing… -3-3 -2-2 -1 0 12 3 4 5 6 7 89 10 1112 ( ) 7 5 they would write… -3-3 -2-2 -1 0 12 3 4 5 6 7 89 10 1112 7 5 next Author’s Preference

28. next © 2007 Herbert I. Gross Prior to this discussion (3,4) referred only to an ordered pair of numbers (or geometrically a point in the xy-plane). However, it should be clear from context whether (3,4) represents an ordered pair or an open interval. Caution

29. A connected interval doesn’t have to be either open or closed. It might be open at one end and closed at the other end. For example, the set {x: a < x ≤ b} would be represented by the notation (a, b]; and it would be referred to as being open at one end and closed at the other end. next © 2007 Herbert I. Gross More Mathematical Notation

30. R = {x: 5 < x ≤ 7} =(5,7] next © 2007 Herbert I. Gross R -3-3 -2-2 -1 0 12 3 4 5 6 7 89 10 1112 ( ] 7 5 Geometric Illustration of a “Mixed” Interval -3-3 -2-2 -1 0 12 3 4 5 6 7 89 10 1112 7 5 next or…

31. © 2007 Herbert I. Gross Real Life Application As an example of intervals that are part open and part closed, consider a mail service that charges $3 to deliver packages that weigh no more than 1 pound; $5 for packages that weigh more than a pound but no more than 2 pounds; and $6 for packages that weigh more than 2 pounds, but no more than 3 pounds.

32. next © 2007 Herbert I. Gross In the language of functions, if we let x represent the weight of the packages in pounds and f(x) the cost in dollars, f would be defined in the following way… f(x) = 3 if x є (0,1] 5 if x є (1,2] 6 if x є (2,3] Notice that f as defined above ensures that for each x in the interval (0,3]; there is one and only one value of f(x). For example, f(2) = 5 not 6 because 2 є (1,2] but 2 є (2,3]. next

33. © 2007 Herbert I. Gross 1 next 234 5 1 6 2 3 4 5 6 Pounds DollarsDollars 0 ( ] ( ] ( ] next In terms of a graph…

34. Given two sets A and B, we define a new set called the union of A and B and written as A U B to be the set of all elements that belong to either A or B. In mathematics the phrase “either...or” is used to mean “at least one”. It does not mean “one or the other but not both". © 2007 Herbert I. Gross In this context… A U B = {x:x є A or x є B} next Unions and Intersections of Sets

35. next © 2007 Herbert I. Gross For example: Then... A U B = { }. 1, 2,3,4,7, 8 If A = {1, 2, 3, 7} and B = {1, 4, 7, 8}

36. next © 2007 Herbert I. Gross Note on Union Even though 1and 7 belong to both A and B, they are only counted once as members of A U B. As a realistic application, suppose we want to know the total number of students who take either math or chemistry. If a student takes both math and chemistry, we only count that person once in the total.

37. next © 2007 Herbert I. Gross next Thus with respect to the previous example, we see that… 1, 7 7 A = {1, 2, 3, 7} and B = {1, 4, 7, 8} A ∩ B = { } A companion to the union of A and B is the intersection of A and B, written A ∩ B. The intersection of A and B consists of those elements that are members of both A and B.

38. we may view the domain of f to be the union of the three intervals (0,1], (1,2] and (2,3]. That is, dom f = (0,1] U (1,2] U (2,3]. next © 2007 Herbert I. Gross So with respect to: f(x) = 3 if x є (0,1] 5 if x є (1,2] 6 if x є (2,3]

39. f 1 (x) = 3, domain of f 1 = (0,1] f 2 (x) = 5, domain of f 2 = (1,2] f 3 (x) = 6, domain of f 3 = (2,3] next © 2007 Herbert I. Gross Moreover, we can then think of f as being the union of the three functions…

40. next © 2007 Herbert I. Gross next In terms of our previous graph… 1 234 5 1 6 2 3 4 5 6 Pounds DollarsDollars 0 ( ] ( ] ( ] f 1 (x) f 2 (x) f 3 (x)

41. For example, suppose that… h(x) = 2x, domain h = [1, 2] and k(x) = 2x, domain k = [3, 4] next © 2007 Herbert I. Gross The important point is that a function is defined not just by its rule but also by its domain. Then even though h and k represent the same rule, they are not equal functions because they have different domains. For example, 3 є dom k but 3 є dom h. next

42. © 2007 Herbert I. Gross next In terms of a graph… y = 2x [ ] y = h(x) 1 234 5 1 6 2 3 4 5 6 0 7 8 [ ] y = k(x) next

43. Given two functions f and g, we say that f = g if and only if… (1) domain of f = domain of g and (2) f(x) = g(x) for each x є dom f (or dom g) © 2007 Herbert I. Gross This leads to the definition… Definition D

44. To indicate that the interval consists of all the non negative numbers, we use the Infinity (∞) symbol and write… domain of f = [0, ∞) next © 2007 Herbert I. Gross “Infinite” Intervals Sometimes a connected interval goes on forever. For example, when we say that the relationship between the area (A) of a square and the length (L) of one of its sides is given by: A = f(L), where f(L) = L 2, it is understood that the domain of f is the set of all non negative numbers.

45. Infinity Is Not a Number There is a tendency for students to think of infinity as “the biggest number” (which might cause a problem with defining ∞ + 1). Rather we should think of infinity as being associated with “Increasing without bound”. In this context, [3, ∞) would mean the set of all numbers that begin with 3 and keep increasing without bound. To indicate that the domain consists of all numbers we write (- ∞, ∞). next © 2007 Herbert I. Gross

46. next Important Question Why couldn't we leave well enough alone and just write P = 4s with out introducing the language of functions? Answer #1: There are times when we know that there is a relationship between 2 quantities, say x and y; but we don't know what the relationship is. We would then write something like y = f(x) to indicate that a relationship exists. next

47. © 2007 Herbert I. Gross next Answer #2: Another answer to this question is that mathematicians, while they have no objection to the practical side of mathematics, prefer to see all mathematical concepts defined in ways that are independent of any one particular practical application. That is, there are many other practical applications in which we multiply the input by 4 to find the output.

48. © 2007 Herbert I. Gross next The cost C, in dollars, of p pounds of candy at a price of $4 per pound is given by C = 4p Example To cover all of the above cases and more, a mathematician might prefer to write f(x) = 4x. This frees the process of multiplying by 4 from any particular application. That is, the expression f(x) = 4x tells us that f is a rule that assigns to any input four times its value. An object traveling at a speed of 4 miles per hour travels M miles in h hours according to the formula M = 4h

49. © 2007 Herbert I. Gross next The use of the letter x is irrelevant; namely f is the rule which given any number, multiplies it by 4. If we had used n to denote the number, we would have written f(n) = 4n. Note next A good way not to become confused by what is the input and what is the output, is to replace x by ( ). That is, think of f(x) = 4x as having been written in the form: f( ) = 4( ). Then we simply make sure that whatever expression we place in one set of parentheses is also placed in the other set.

50. © 2007 Herbert I. Gross next f( ) = 4( ) = Thus, for example… So with respect to the previous examples we could write… P = f(s) = 4s C = f(p) = 4p M = f(h) = 4h 28 7 7 7 7 f( ) = 4( ) = - 16 -4-4 -4-4 -4-4 -4-4 f( ) = 4( ) = 4t + 4 t + 1

51. © 2007 Herbert I. Gross next This concludes our present discussion on functions. In part 3 of this lesson we will discuss graphs in more detail.