Mathematics

1. Mathematics

2. Session Set, Relation & Function Session - 3

3. Session Objectives

4. Session Objectives • Function definition • Domain, codomain and range • Standard real functions • Types of functions • Number of various types of functions • Composition of functions • Inverse of element • Inverse of function

5. Function Let A and B be two non-empty sets having m and n elements respectively, then the number of relations possible from A to B is 2mn. Out of these 2mn relations some are called function (or mappings) from A to B provided following two conditions hold in the relation: • All the elements of A are associated to elements of B. • Each element of A is associated to one and only one element of B, i.e. no element of A is associated to two (or more) elements of B.

6. Function ‘f’ from set A to set B associateseach element of A to unique (i.e. one andonly one) element of B denoted by(read as ‘f from A to B’) Definition Observations: (i)A relation from A to B is not a function if it either violates condition 1 or 2 or both, i.e. either some element of A is not associated to element of B or some element of A is associated to more than one elements of B or both. (ii)In a function from A to B, two elements of A can be associated to one element of B (examples R7, R10)

7. (iii)If be the function, theno(f) = o(A) and Dom(f) = A. Observations (i)A relation from A to B is not a function if it either violates condition 1 or 2 or both, i.e. either some element of A is not associated to element of B or some element of A is associated to more than one elements of B or both. (ii)In a function from A to B, two elements of A can be associated to one element of B (examples R7, R10)

8. Let be the function, then set ‘A’is called the domain of f and set ‘B’ iscalled thecodomain of f. The set ofthose elements of B which are relatedby elements of A is called rangeof for image of set A under f and isdenoted by f(A), i.e.Range of f. Clearly, Domain, Codomain and Range of a Function

9. Dom (R7) = {a, b},Codomain = {1, 2} Range (R7) = {1} Domain (R8) = {a, b} Codomain (R8) = {1, 2} Range (R8) = {1, 2} = Codomain (R8) Domain, Codomain and Range of a Function For example:

10. Dom (R7) = {a, b},Codomain = {1, 2} Range (R7) = {1} Domain (R8) = {a, b} Codomain (R8) = {1, 2} Range (R8) = {1, 2} = Codomain (R8) Domain, Codomain and Range of a Function For example:

11. (iii)or Dom (g) Equal Functions Two functions f and g are said to be equal iff (i)Dom (f) = Dom (g) (ii)Codom (f) = Codom (g) If all these three conditions holds, thenwe can write f = g.

12. If A and B be two non-empty sets,f be the relation from A to B (i.e. ),then f is function from A to B if (i) and (ii) Mathematical Way to Prove a Relation to be a Function

13. Some Standard Real Functions and Their Graphs Real functions Functions in which both domain and codomain are the subsets of R, i.e. set of real numbers. Domain of f is [a, c] Codomain is R Range is [u, w]

14. Let f : R R be the real function defined as Note: f : A B is constantfunction if f(a) = cfor some Some Standard Real Functions and Their Graphs Constant function: f(x) = c Dom (f) = R, Codomain (f) = R,Range (f) = {c}

15. Let f : R R be the real function defined as Note: f : A A given by f(a) = a is identity function denotedby (same as identity relation) Some Standard Real Functions and Their Graphs Identity function: f(x) = x Dom (f) = RCodomain (f) = RRange (f) = R

16. Let f : R R be the real function defined as Range (f) = (as ) = set of non-negative real numbers. Some Standard Real Functions and Their Graphs Modulus function: f(x) = |x| Domain (f) = R, Codomain (f) = R,

17. Let f : R R be the real function defined as = greatest integer less than or equal to x. Some Standard Real Functions and Their Graphs Greatest integer function: f(x) = [x] For example: [2.1] = 2, i.e. greatest integer less than or equal to 2.1 is 2, similarly [–2.1] = –3 [2] = 2 [3 . 9] = 3 [–3 . 9] = –4

18. Hence [x] = 0 = 1 = 2 and so on. Also [x] = –1 = –2 and so on. Combining we get [x] = n for Some Standard Real Functions and Their Graphs

19. Some Standard Real Functions and Their Graphs Filled circle means, point is on the graph. unfilled circle means, point is not on the graph. Dom(f) = R, Codomain (f) = R Range (f) = z

20. Let f : R R be the real function defined as Say , then Some Standard Real Functions and Their Graphs Exponential function: f(x) = ax Case I: Let 0 < a < 1

21. Some Standard Real Functions and Their Graphs Case II:Let a > 1 Say a = 2, then y = ax = 2x

22. Domain (f) = R, Codomain (f) = R,Range (f) = Some Standard Real Functions and Their Graphs Special case a = e > 1 y = f(x) = ex

23. Let be the real function defined as Some Standard Real Functions and Their Graphs Logarithmic function: y = logbx Case 1: 0 < b < 1

24. Some Standard Real Functions and Their Graphs Case 2: 1 < b Domain (f) = Codomain (f) = R Range (f) = R

25. A function is said to be one-onefunction or injective if different elementsof A have different images in B, i.e. if Thus iff Types of Functions One-one function (or injective)

26. For example: Let be thefunction given by (i) (ii) (iv) (iii) Types of Functions

27. In (i) In (iv) (ii)is true for all the functions (condition 2)but its converse is true forone-one function. Types of Functions Here only (ii) and (iii) are one-one functions Hence (i) and (iv) are not one-one functions. Observation: (i)To check injectivity of functionsLet f(x) = f(y) if it gives x = y only, then f is a one-one function.

28. (iv)If Types of Functions (iii)Injectivity of f(x) can also bechecked by itsgraph. If all lines parallel to x-axis cut f(x) innotmore than one point, then f(x) is one-onefunction, i.e. f(x) is not a one-one function ifat leastone line parallel to x-axis cuts f(x)in more than one point.

29. Alternative: f(1) = f(–1) = 1 but (i) Types of Functions Using graphs Clearly, any line parallel to x-axis cuts f(x) = 2xonly at one point, thus f(x) is one-one.

30. (ii) (iii) Types of Functions Clearly, y = 4 cuts y = x2 intwo points hence f(x) = x2is not one-one Clearly, y = 1 cuts y = |x|in two points hence f(x) = |x|is not one-one.

31. A function which is not one-oneis many-one function, i.e. at least twodifferent elements of A have sameimage in B or s.t. Butf(x) = f(y). For example:given by , are both many-one functions as . Types of Functions Many-one function

32. A function is said to be ontofunction or subjective if all theelements of B have preimage in A,i.e. for each i.e. A function is not onto if s.t. thereis no for which f(a) = b. Types of Functions Onto function (or surjective)

33. Let be the function given by (ii) (i) (iii) (iv) Types of Functions For example:

34. In (i) have no pre-image in A and in(iv) have no pre-image in A, thus notonto functions. (i) is onto function if Range (f) = Codomain (f) = B Proof:(by definition) let ,then if f is onto it has pre-image in A (As range (f) contains those elementsof B which have pre-image in A). Types of Functions Here functions (ii) and (iii) are onto functions as all the elements of B have pre-image in A. Observations:

35. Range (f) = Codomain (f) (ii)To check surjectivity of function takesomeand follow the steps given below: (iii)If is onto, then Types of Functions Hence codomain (f) Step 1: Let f(x) = y Step 2: Express x in terms of y from above equation say x = g(y) Step 3: Now find the domain of g, ifDom (g) = Codomain (f) (i.e. B) then f is onto (or surjective)

36. A function which is not onto is into function,i.e. at least one element of B have no pre-image in A or such that there is no for which f(a) = b. Types of Functions Into function So in example given above, functions, (i) and(iv) are into functions. Bijective function (or one-one and onto) A function is said to be bijective if it isinjective as well as surjective, i.e.one-one as well as onto.

37. In other words is bijective if (i)f is one-one, i.e. and (ii)f is onto, i.e. for each somest f(a) = b or Types of Functions Range (f) = Codomain (f)

38. (ii)If f is bijective, then it is injective and surjective Types of Functions Observations: (i)A function is not bijective if it is eithernot injective or not surjective or notboth. Hence o(A) = o(B).

39. Let A and B be two non-empty sets, leto(A) = m, o(B) = n and be thefunction from A to B Each element of A can be associated to nelements of B, so total number of functionsthat can be formed from A to B isn × n × ... × n (m times), i.e. nm. Hencetotal number of functions fromA to B = Number of Functions of Various Types Number of functions from A to B

40. (iii)Number of relations from A to B which arenot functions is 2mn – nm or . Number of Functions of Various Types Number of functions from A to B • Every function is a relation but not vice versa. • If A and B are two non-empty sets such that o(A) = m, o(B) = n, then number of relationspossible from A to B is 2mn and number of functions possible from A to B is nm.

41. Out of nm functions, from A to B some are one-one functions. Now if we order the elements of A from 1 to m say first, second, ..., mth then for first element of A we have n choices from set B, for second we have (n – 1) choices from set B (as function has to be one-one) and so on. Thus total number of one-one functions possible from A to B is n (n – 1) (n – 2) ... (n – m + 1), i.e. Here note that m has to be less than or equal to n, i.e. otherwise if m is greater than n, i.e. no 1-1 function is possible from A to B (as in that case first n elements of A will be associated to n elements of B and still m – n elements of A remains to be associated). Hence number of 1-1 functions from A to B. Number of Functions of Various Types Number of one-one functions from A to B

42. (i) Out of functions from A to B, functions are one-one (provided ). (ii)If functions from A to Bare many-one functions. Number of Functions of Various Types Observations: (iii)If m > n, then all the nmfunctions aremany-one functions.

43. Number of Functions of Various Types Number of bijective functionsfrom A to B A function is bijective iff function is 1-1 as well as onto. This implies o(A) = o(B),i.e. m = n. Hence for first element of A we have n options, for second we have (n – 1) options and so on, for last we have only one option. Therefore, total number of bijective functions from A to B is n (n – 1) ... 2.1 = n!

44. Show all bijective functions fromwhere A = {a, b, c} and B = {x, y, z} Hence number of bijective functions Number of Functions of Various Types

45. Number of Functions of Various Types Observations: (i) If o(A) = o(B) and function is 1-1, then function is onto also and hence bijective. (ii) If o(A) = o(B) and function is onto, then function is 1-1 also and hence bijective.

46. Let be two functions, then function gof : defined as is called the composition of f and g Composition of Functions Observations: (i)For gof to exist, range of f must be subset of domain of g. (ii)Similarly for fog to exit, range of g must be subset of domain of f.

47. (i)Composition of functions is non commutative, i.e.Note that its possible that fog may not exist even if gof exists. (iv)Composition of identity function with anyfunction f : is f itself, i.e. . Composition of Functions Properties of composition offunctions • Composition of functions is associative, i.e.f, g, h be three functions.Then (fog) oh = fo (goh) (provided they exist) • If f and g are bijections, then gof is also a bijection (provided exist)

48. Let be the function fromA to B, then for any if f(a) = b,then a is called the pre-image of ‘b’or inverse of ‘b’ denoted by f–1(b). For example: Let be given by, Inverse of Element Note:Inverse of an element maynot be unique.

49. If be a bijective function,then we can define a new functionfrom B to A as inverse of f denotedby given by Inverse of Function i.e. each element of B is associated(or mapped) to its pre-image under f

50. Ifis a bijective function, thenf–1 can be obtained using following steps: (iii)Interchanging x with y, i.e. , we gety = g(x). Then g = f–1 Inverse of Function How to find f–1 (i)Let y = f(x) (ii)Express x in terms of y, say x = g(y) Note that before finding f–1, you have to provethat f is a bijective function (i.e. 1-1 as well asonto) by using the rules given before.