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## Mathematics

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**Session**Set, Relation & Function Session - 3**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**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.**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)**(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)**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**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:**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:**(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.**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**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]**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}**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**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,**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**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**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**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**Some Standard Real Functions and Their Graphs**Case II:Let a > 1 Say a = 2, then y = ax = 2x**Domain (f) = R, Codomain (f) = R,Range (f) =**Some Standard Real Functions and Their Graphs Special case a = e > 1 y = f(x) = ex**Let be the real function defined as**Some Standard Real Functions and Their Graphs Logarithmic function: y = logbx Case 1: 0 < b < 1**Some Standard Real Functions and Their Graphs**Case 2: 1 < b Domain (f) = Codomain (f) = R Range (f) = R**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)**For example: Let be thefunction given by**(i) (ii) (iv) (iii) Types of Functions**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.**(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.**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.**(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.**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**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)**Let be the function given by**(ii) (i) (iii) (iv) Types of Functions For example:**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:**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)**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.**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)**(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).**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**(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.**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**(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.**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!**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**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.**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.**(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)**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.**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**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.