Relations -I MDULE - I RELATINS AND FUNCTINS-I In our daily life, we come across many patterns that characterise relations such as brother and sister, father and son, teacher and student etc. In mathematics also, we come across many relations such as number m is greater than number n, line is perpendicular to line m etc. the concept of relation has been established in mathematical form. The word function was introduced by leibnitz in 694. Function is defiend as a special type of relation. In the present chapter we shall discuss cartesion product of sets, relation between two sets, conditions for a relation to be a function, different types of functions and their properties. BJECTIVES After studying this lesson, you will be able to : define cartesion product of two sets. define. relation, function and cite eamples there of find domain and range of a function draw graph of functions. define and cite eamples of even and odd turnotions. determine whether a function is odd or even or neither define and cite eamples of functions like, polynomial functions, logarithmic and eponential functions the greatest integer functions, to Find sum. difference, product and quotient of real functions. EXPECTED BACKGRUND KNWLEDGE concept of ordered pairs.. CARTESIAN PRDUCT F TW SETS Consider two sets A and B where A={, }, B= {3, 4, 5}. Set of all ordered pairs of elements of A and B MATHEMATICS 5
Relations -I MDULE - I is {(,3), (,4), (,5), (,3), (,4), (,5)} This set is denoted by A B and is called the cartesian product of sets A and B. i.e. A B ={(, 3), (, 4),(, 5),(, 3),(, 4),(, 5)} Cartesian product of sets B and A is denoted by B A. In the present eample, it is given by B A = {(3, ),(3, ),(4, ),(4, ),(5, ),(5, )}, Clearly A B B A. In the set builder form : A B = {(a,b) : a A and b B} and B A = {(b,a) : b B and a A } Note : If A or B or A, B then A B B A. Eample. () Let A={a,b,c}, B={d,e}, C={a,d}. Solution : Find (i) A B(ii) B A (iii) A ( B C) (iv)(a C) B (v)(a B) C (vi) A (B C). (i) A B ={(a, d),(a, e), (b, d), (b, e), (c, d), (c, e)}. (ii) B A = {(d, a),(d, b), (d, c), (e, a) (e, b),(e, c)}. (iii) A = {a, b, c}, B C ={a,d,e}. A ( B C ) ={(a, a),(a, d),(a, e),(b, a),(b, d),(b, e), (c, a),(c, d),(c, e). (iv) A C = {a}, B={d, e}. ( A C ) B={(a, d), (a, e)} (v) A B =, C={a,d}, A B C (vi) A = {a,b,c}, B C {e}. A (B C) {(a,e),(b,e),(c,e)}... Number of elements in the Cartesian product of two finite sets Let A and B be two non-empty sets. We know that A B = {(a, b); a A and b B} Then number of elements in Cartesian product of two finite sets A and B i.e. n(a B) = n(a). n(b) Eample. Suppose A = {,,3}and B={, y}, show that n A B n A nb Solution : Here n(a) = 3, n(b) = A B ={(, ), (, ), (3, ), (, y), (, y), (3, y)} n(a B) = n(a) n(b), = 3 = 6 Eample.3 If n(a) = 5, n(b) = 4, find n(a B) 6 MATHEMATICS
Relations -I Solution : We know that n(a B) =n(a) n(b) n(a B) = 5 4 = 0 MDULE - I.. Cartesian product of the set of real numbers R with itself upto R R R rdered triplet A A A = {(a, b, c) : a, b, c A} Here (a, b, c) is called an ordered triplet. The Cartesian product R R represents the set R R = {(, y) :, y R} which represents the coordinates of all the points in two dimensional plane and the Cartesian product R R R represent the set R R R = {(, y, z) :, y, z R} which represents the coordinates of all the points in three dimenstional space. Eample.4 If A = {, }, form the set A A A. Solution : A A A ={(,, ), (,, ), (,, ), (,, ) (,, ), (,, ), (,, ), (,, )}. RELATINS Consider the following eample : A={Mohan, Sohan, David, Karim} and B={Rita, Marry, Fatima} Suppose Rita has two brothers Mohan and Sohan, Marry has one brother David, and Fatima has one brother Karim. If we define a relation R " is a brother of" between the elements of A and B then clearly. Mohan R Rita, Sohan R Rita, David R Marry, Karim R Fatima. After omiting R between two names these can be written in the form of ordered pairs as : (Mohan, Rita), (Sohan, Rita), (David, Marry), (Karima, Fatima). The above information can also be written in the form of a set R of ordered pairs as R= {(Mohan, Rita), (Sohan, Rita), (David, Marry), Karim, Fatima} Clearly R A B, i.e.r {(a,b) :a A, b B and arb} If A and B are two sets then a relation R from A tob is a sub set of A B. If (i) R, R is called a void relation. (ii) R=A B, R is called a universal relation. (iii) If R is a relation defined from A to A, it is called a relation defined on A. (iv) R = (a, a) a A, is called the identity relation... Domain and Range of a Relation If R is a relation between two sets then the set of first elements (components) of all the ordered pairs of R is called Domain and set of nd elements of all the ordered pairs of R is called range, MATHEMATICS 7
Relations -I MDULE - I of the given relation. In the previous eample. Domain = {Mohan, Sohan, David, Karim}, Range = {Rita, Marry, Fatima} Eample.5 Given that A = {, 4, 5, 6, 7}, B = {, 3}. R is a relation from A to B defined by R = {(a, b) : a A, b B and a is divisible by b} find (i) R in the roster form (ii) Domain of R (iii) Range of R (iv) Repersent R diagramatically. Solution : (i) R = {(, ), (4, ), (6, ), (6, 3)} (ii) Domain of R = {, 4, 6} (iii) Range of R = {, 3} (iv) Fig.. Eample.6 If R is a relation 'is greater than' from A to B, where A= {,, 3, 4, 5} and B = {,,6}. Find (i) R in the roster form. (ii) Domain of R (iii) Range of R. Solution : (i) R = {(,), (3, ), (3, ), (4, ), (4, ), (5, ), (5, )} (ii) Domain of R = {, 3, 4, 5} (iii) Range of R = {, }.. Co-domain of a Relation If R is a relation from A to B, then B is called codomain of R. For eample, let A = {, 3, 4, 5, 7} and B = {, 4, 6, 8} and R be the relation is one less than from A to B, then R = {(, ), (3, 4), (5, 6), (7, 8)} so codomain of R = {, 4, 6, 8} Eample.7 : Let A = {,, 3, 4, 5, 6}. Define a relation R from A to A by R = {(, y) : y = + } and write down the domain, range and codomain of R. Solution : R = {(, ), (, 3), (3, 4), (4, 5), (5, 6)} Domain of A = {,, 3, 4, 5} Range of R = {, 3, 4, 5, 6} and Codomain of R = {,, 3, 4, 5, 6} CHECK YUR PRGRESS.. Given that A = {4, 5, 6, 7}, B = {8, 9}, C = {0} Verify that A (B C) = (A B) (A C). 8 MATHEMATICS
Relations -I. If U is a universal set and A, B are its subsets. Where U= {,, 3, 4, 5}. A = {,3,5}, B = { : is a prime number}, find A' B' 3. If A = {4, 6, 8, 0}, B = {, 3, 4, 5}, R is a relation defined from A to B where R= {(a, b) : a A, b B and a is a multiple of b} find (i)r in the Roster form (ii) Domain of R (iii) Range of R. 4. If R be a relation from N to N defined by R= {(,y) : 4 y,, y N } find (i) R in the Roster form (ii) Domain of R (iii) Range of R. MDULE - I 5. If R be a relation on N defined by R={ (, ) : is a prime number less than 5} Find (i) R in the Roster form (ii) Domain of R (iii) Range of R 6. If R be a relation on set of real numbers defined by R={(,y) : y 0 }, Find (i) R in the Roster form (ii) Domain of R (iii) Range of R. 7. If ( +, y ) = (3, ), find the values of and y. 8. If A = {, } find A A A. 9. If A B = {(a, ), (a, y), (b, ), (b, y)}. Find A and B. 0. If n(a) = 6 and n(b) = 3, then find n(a B)..3 DEFINITIN F A FUNCTIN Consider the relation f : {(a,), (b,), (c,3), (d,5)}from set A = {a,b,c,d}to set B = {,,3,4}. In this relation we see that each element of A has a unique image in B. This relation f from set A to B where every element of A has a unique image in B is defined as a function from A to B. So we observe that in a function no two ordered pairs have the same first element. Fig.. We also see that an element B, i.e., 4 which does not have its preimage in A. Thus here: (i) the set B will be termed as co-domain and (ii) the set {,, 3, 5} is called the range. From the above we can conclude that range is a subset of co-domain. Symbolically, this function can be written as f : A B or A f B.3. Real valued function of a real variable 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, then it is called a real function. MATHEMATICS 9
MDULE - I Relations -I Let R be the set of all real numbers and X, Y be two non-empty subsets of R, then a rule f which associates to each X, a unique element y of Y is called a real valued function of the real variable or simply a real function and we write it as f : X Y A real function f is a rule which associates to each possible real number, a unique real number f(). Eample.8 Which of the following relations are functions from A to B. Write their domain and range. If it is not a function give reason? (a) (, ), (3, 7), (4, 6), (8,), A, 3, 4, 8, B, 7, 6,, (b) (, 0), ( ), (, 3), (4,0), A,, 4, B 0,, 3,0 (c) (a, b), (b, c), (c, b), (d, c), A a, b, c, d, e B b, c (d) (, 4), (3, 9), (4,6), (5, 5), (6, 36, A, 3, 4, 5, 6, B 4, 9,6, 5, 36 (e) (, ), (, ), (3, 3), (4, 4), (5, 5), A 0,,, 3, 4, 5 (f) B,, 3, 4, 5, 3 sin,, cos,, tan,, cot, 3 6 6 6 3 6, A sin, cos, tan, cot 6 6 6 6 3 B,,, 3, 3 (g) (a, b), (a, ), (b, 3), (b, 4), Solution : A a, b, B b,, 3, 4. (a) It is a function. Domain =, 3, 4, 8, Range, 7, 6, (b) It is not a function. Because Ist two ordered pairs have same first elements. (c) It is not a function. Because Domain = a, b, c, d A, Range = b, c (d) It is a function. Domain, 3, 4, 5, 6, Range 4, 9,6, 5, 36 (e) It is not a function. Because Domain,, 3, 4, 5 A, Range,, 3, 4, 5 (f) It is a function. Domain sin, cos, tan, cot 6 6 6 6 3, Range,,, 3 3 30 MATHEMATICS
Relations -I (g) It is not a function. Because first two ordered pairs have same first component and last two ordered pairs have also same first component. Eample.9 State whether each of the following relations represent a function or not. MDULE - I (a) (b) Fig..3 Fig..4 (c) (d) Solution : Fig..5 Fig..6 (a) f is not a function because the element b of A does not have an image in B. (b) f is not a function because the element c of A does not have a unique image in B. (c) f is a function because every element of A has a unique image in B. (d) f is a function because every element in A has a unique image in B. Eample.0 Which of the following relations from R. R are functions? (a) y 3 (b) y 3 (c) y Solution : (a) y 3 Here corresponding to every element R, a unique element y R. It is a function. (b) y 3. For any real value of we get more than one real value of y. It is not a function. (c) y For any real value of, we will get a unique real value of y. It is a function. MATHEMATICS 3
MDULE - I Relations -I Eample. : Let N be the set of natural numbers. Define a real valued function f : N N by f() = +. Using the definition find f(), f(), f(3), f(4) Solution : f() = + f() = + = + = 3, f() = + = 4 + = 5 f(3) = 3 + = 6 + = 7, f(4) = 4 + = 6 + = 9 CHECK YUR PRGRESS.. Which of the following relations are functions from A to B? (a) (, ), (3, 7), (4, 6), (8,), A,3, 4,8, B, 7, 6, (b) (, 0), (, ), (, 3), (4,0), (c) A,, 4, B, 0,, 3,0 (a, ), (b, 3), (c, ), (d, 3), A a, b, c, d, B, 3 (d) (,), (, ), (, 3), ( 3, 4), (e),, 3,,..., 0,, 3 0 A,, 3, B,, 3, 4 A,, 3, 4, B,,..., 3 (f),,,,, 4,, 4, A 0,,,,, B, 4. Which of the following relations represent a function? (a) (b) Fig..7 Fig..8 (c) (d) Fig..9 Fig..0 3 MATHEMATICS
Relations -I 3. Which of the following relations defined from R R are functions? (a) y (b) y 3 (c) y 3 (d) y 4. Write domain and range for each of the following functions : (a),, 5,, 3, 5 (c),, 0, 0,,,,, (b) 3,,,,, (d) Deepak,6, Sandeep, 8, Rajan, 4 5. Write domain and range for each of the following mappings : (a) (b) MDULE - I Fig.. Fig.. (c) (d) Fig..3 Fig..4 (e) Fig..5 MATHEMATICS 33
MDULE - I.3. Some More Eamples on Domain and Range Relations -I Let us consider some functions which are only defined for a certain subset of the set of real numbers. Eample. Find the domain of each of the following functions : (a) y (b) y (c) y ( )( 3) Solution : The function y can be described by the following set of ordered pairs....,,,,,,,,... Here we can see that can take all real values ecept 0 because the corresponding image, i.e., 0 is not defined. Domain R 0 Note : Here range = R{ 0 } [Set of all real numbers ecept 0] (b) can take all real values ecept because the corresponding image, i.e., not eist. Domain R (c) Value of y does not eist for and 3 Domain R, 3 does Eample.3 Find domain of each of the following functions : (a) y (b) y 4 Solution :(a) Consider the function y In order to have real values of y, we must have Domain of the function will be all real numbers. (b) y 4 0 i.e. In order to have real values of y, we must have 4 0 We can achieve this in the following two cases : Case I : 0 and 4 0 and 4 Domain consists of all real values of such that 4 Case II : 0 and 4 0 and 4. But, cannot take any real value which is greater than or equal to and less than or equal to 4. 34 MATHEMATICS
Relations -I From both the cases, we have Domain = 4 R Eample.4 For the function find the range when domain 3,,, 0,,, 3 f y, Solution : For the given values of, we have f 3 3 5, f 3 f, f 0 0., f 5, f 3 The given function can also be written as a set of ordered pairs. f 3 3 7 i.e., 3, 5,, 3,,, 0,, 3,, 5 3, 7 Range 5, 3,,, 3, 5, 7 Eample.5 If f 3, 0 4,find its range. Solution : Here 0 4 or 0 3 3 4 3 Range = f : 3 f 7 or 3 f 7 MDULE - I Eample. 6 If f, 3 3,find its range. Solution : Given 3 3 or 0 9 Range = f : 0 f 9 or 0 f 9 CHECK YUR PRGRESS.3. Find the domain of each of the following functions R : (a) (i) y =, (ii) y = 9 + 3, (iii) y 5 (b) (i) y 3 y, (ii) 4 5 (iii) y 3 5, (iv) y 3 5 (c) (i) y 6, (ii) y 7, (iii) y 3 5 (d) (i) y 3 5 (ii) y 3 5 MATHEMATICS 35
MDULE - I (iii) y 3 7 (iv) y Relations -I 3 7. Find the range of the function, given its domain in each of the following cases. (a) (i) f 3 0,, 5, 7,,, (ii) f, 3,, 4, 0 (iii) f,,, 3, 4, 5 (b) (i) f, 0 4 (ii) f 3 4, (c) (i) f, 5 5 (ii) f, 3 3 (iii) f, (iv) f, 0 5 (d) (i) f 5, R (ii) f 3, R (iii) f 3, R (iv) f (v) f (vii) f, : 0, : (vi) f 3, : 0, : 0 (viii) f 5.4 GRAPHICAL REPRESENTATIN F FUNCTINS, : 5 Since any function can be represented by ordered pairs, therefore, a graphical representation of the function is always possible. For eample, consider y. 0 3 3 4 4 y 0 4 4 9 9 6 6 Fig..6 36 MATHEMATICS
Relations -I Does this represent a function? Yes, this represent a function because corresponding to each value of a unique value of y. Now consider the equation y 5 0 0 3 3 4 4 5 5 3 3 4 4 y 5 5 4 4 3 3 0 0 4 4 3 3 MDULE - I Fig..7 This graph represents a circle. Does it represent a function? No, this does not represent a function because corresponding to the same value of, there does not eist a unique value of y. CHECK YUR PRGRESS.4. (i) Does the graph represent a function? Fig..8 MATHEMATICS 37
Relations -I MDULE - I (ii) Does the graph represent a function? Fig..9. Draw the graph of each of the following functions : (a) y 3 (b) y (c) y (d) y 5 (e) y (f) y 3. Which of the following graphs represents a function? (a) (b) Fig..0 Fig.. (c) (d) Fig.. Fig..3 38 MATHEMATICS
Relations -I (e) MDULE - I Fig..4 Hint : If any line to y-ais cuts the graph in more than one point, graph does not represent a function..5 SME SPECIAL FUNCTINS.5. Monotonic Function Let F : A B be a function then F is said to be monotonic on an interval (a,b) if it is either increasing or decreasing on that interval. For function to be increasing on an interval (a,b) F F a, b and for function to be decreasing on (a,b) F F a, b A function may not be monotonic on the whole domain but it can be on different intervals of the domain. Consider the function F : R R defined by f. Now, 0,, F F F is a Monotonic Function on 0,. ( It is only increasing function on this interval) But,, 0, F F F is a Monotonic Function on, 0 ( It is only a decreasing function on this interval) Therefore if we talk of the whole domain given function is not monotonic on R but it is monotonic on, 0 and 0,. Again consider the function F : R R defined by f 3 Clearly domain F F Given function is monotonic on R i.e. on the whole domain.. MATHEMATICS 39
MDULE - I.5. Even Function Relations -I A function is said to be an even function if for each of domain F( ) F() For eample, each of the following is an even function. (i) If F then F F (ii) If F (iii) If F cos then F cos cos F then F F Fig..5 The graph of this even function (modulus function) is shown in the figure above. bservation Graph is symmetrical about y-ais..5.3 dd Function A function is said to be an odd function if for each For eample, f f (i) If f 3 3 3 then f f (ii) If f sin then f sin sin f Graph of the odd function y = is given in Fig..6 Fig..6 bservation Graph is symmetrical about origin..5.4 Greatest Integer Function (Step Function) f which is the greatest integer less than or equal to. 40 MATHEMATICS
Relations -I f is called Greatest Integer Function or Step Function. Its graph is in the form of steps, as shown in Fig..7. Let us draw the graph of,, 3 3, 3 4 0, 0 y, R, 0 MDULE - I, Domain of the step function is the set of real numbers. Range of the step function is the set of integers. Fig..7.5.5 Polynomial Function Any function defined in the form of a polynomial is called a polynomial function. For eample, (i) f 3 4, (ii) f 3 5 5, (iii) are all polynomial functions. f 3 Note : Functions of the type f k, where k is a constant is also called a constant function..5.6 Rational Function Function of the type f functions are called rational functions. For eample, g h, where h 0 and g and 4 f, is a rational function. h are polynomial.5.7 Reciprocal Function: Functions of the type y, 0 is called a reciprocal function..5.8 Eponential Function A swiss mathematician Leonhard Euler introduced a number e in the form of an infinite series. In fact e...... 3 n...() It is well known that the sum of this infinite series tends to a finite limit (i.e., this series is convergent) and hence it is a positive real number denoted by e. This number e is a transcendental irrational number and its value lies between an 3. MATHEMATICS 4
MDULE - I 3 n Now consider the infinite series...... 3 n Relations -I It can be shown that the sum of this infinite series also tends to a finite limit, which we denote by e. Thus, 3 n e...... 3 n...() This is called the Eponential Theorem and the infinite series is called the eponential series. We easily see that we would get () by putting = in (). The function f e, where is any real number is called an Eponential Function. The graph of the eponential function y e is obtained by considering the following important facts : (i) As increases, the y values are increasing very rapidly, whereas as decreases, the y values are getting closer and closer to zero. (ii) There is no -intercept, since e 0 for any value of. (iii) The y intercept is, since e0 and e 0. (iv) The specific points given in the table will serve as guidelines to sketch the graph of e (Fig..8). 3 0 3 y e 0.04 0.3 0.36.00.7 7.38 0.08 Fig..8 4 MATHEMATICS
Relations -I If we take the base different from e, say a, we would get eponential function f a,provbided a 0, a. For eample, we may take a = or a = 3 and get the graphs of the functions MDULE - I and y (See Fig..9) y 3 (See Fig..30) Fig..9 Fig..30 Fig..3 MATHEMATICS 43
MDULE - I.5.9 Logarithmic Functions Now Consider the function y e...(3) We write it equivalently as is the inverse function of y e loge y Thus, y e The base of the logarithm is not written if it is e and so log...(4) Relations -I loge is usually written as log. As y e and y log are inverse functions, their graphs are also symmetric w.r.t. the line y. The graph of the function y log can be obtained from that of y e by reflecting it in the line y =. Fig..3 Note (i) The learner may recall the laws of indices which you have already studied in the Secondary Mathematics : If a > 0, and m and n are any rational numbers, then (ii) am an am n, am an am n, am n The corresponding laws of logarithms are amn, 0 a m log a mn loga m loga n, log a loga m loga n n log mn n log m, a Here a, b > 0, a, b. a log m or log b m log a m log b a log a b m log a b 44 MATHEMATICS
Relations -I.5.0 Identity Function Let R be the set of real numbers. Define the real valued functionf:rrby y = f() = for each R. Such a function is called the identity function. Here the domain and range of f are R. The graph is a straight line. It passes through the origin. MDULE - I.5. Constant Function Fig..33 Define the function f : R R by y = f() = c, R where c is a constant and each R. Here domain of f is R and its range is {c}. The graph is a line parallel to -ais. For eample, f() = 4 for each R, then its graph will be shown as y 6 4 6 4 4 6 4 6 y f( ) = 4 Fig..34 MATHEMATICS 45
MDULE - I Relations -I.5. Signum Function if 0 The function f : R R defined by f() = 0, if 0 is called a signum function.,if 0 The domain of the signum function is R and the range is the set {, 0, }. The graph of the signum function is given as under : Fig..35 CHECK YUR PRGRESS.5. Which of the following statements are true or false. (i) function f 4 7 9 is an even function. (ii) dd function is symmetrical about y-ais. (iii) f / 3 5 is a polynomial function. (iv) f (v) f 3 is a rational function for all R. 3 5 is a constant function. 3 (vi) Domain of the function defined by f (vii) Greatest integer function is neither even nor odd.. Which of the following functions are even or odd functions? is the set of real numbers ecept 0. (a) f (b) f 5 (c) f 5 46 MATHEMATICS
Relations -I (d) f (e) f 3 (f) f 5 5 MDULE - I (g) f 3 3 (h) 3 f 3. Draw the graph of the function y. 4. Specify the following functions as polynomial function, rational function, reciprocal function or constant function. (a) y 38 57 85 (b) (c) (e) (g) 3 y, 0 (d) y, 0 (f) y. 9 y 3 3, 3 3 0 y 3, 0 5 6 y,.6 Sum, difference, product and quotient of functions (i) Addition of two real functions : Let f : X R and g : X R be any two functions, where X R. Then, we define (f + g) : X R by (f + g)() =f() + g(), for all X Let f() =, g() = + Then (f + g) () = f() + g() = + + (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) = f() g(), for all X Let f() =, g() = + then (f g) () = f() g() = ( + ) = (iii) Multiplication of two real functions : The product of two real functions f : X R and g : X R is a function f g : X R defined by (f g) () = f(). g(), for all X Let f() =, g() = + Then f g() = f(). g() =. ( + ) = 3 + MATHEMATICS 47
MDULE - I (iv) Quotient of two real functions : Relations -I Let f and g be two real functions defined from X R where X R. The real quotient of f by g denoted by f g is a function defined by f g f = ( ), provided g() 0, X g( ) Let f ()=, g() = + f Then ( ) g = f ( ), g( ) Eample. 7 Let f ( ) and g() = be two functions defined over the set of f non-negative real numbers. Find (f + g)(), (f g)(), (f g)() and ( ) g. Solution : We have f ( ), g( ) Then(f + g)() = f() + g() = (f g)() = f() g() = (f g)() =f(). g() = 3. f ( ) g f ( ) =, 0 g( ) CHECK YUR PRGRESS.6. A function f is defined as f() = 3 + 4. Write down the values of (i) f(0) (ii) f(7) (iii) f( 3). Let f, g : R R be defined, respectively by f() = +, g() = 3. Find (f + g), (f g) (f g) and f g. A C % + LET US SUM UP Cartesian product of two sets A and B is the set of all ordered pairs of the elements of A and B. It is denoted by A B i.e A B ={(a,b): a Aand b B } 48 MATHEMATICS
Relations -I Relation is a sub set of A B where A and B are sets. i.e. R A B a, b : a A and b B and arb Function is a special type of relation. Functrions f : A B is a rule of correspondence from A to B such that to every element of A a unique element in B. Functions can be described as a set of ordered pairs. Let f be a function from A to B. Domain : Set of all first elements of ordered pairs belonging to f. Range : Set of all second elements of ordered pairs belonging to f. Functions can be written in the form of equations such as y f () where is independent variable, y is dependent variable. Domain : Set of independent variable. Range : Set of dependent variable. Every equation does not represent a function. Vertical line test : To check whether a graph is a function or not, we draw a line parallel to y-ais. If this line cuts the graph in more than one point, we say that graph does not represent a function. A function is said to be monotonic on an interval if it is either increasing or decreasing on that interval. A function is called even function if f f, and odd function if f f,, Df f, g : X R and X R, then (f + g)() = f() + g(), (f g)() = f() g() MDULE - I (f. g) = f(). g(), f ( ) g f ( ) g( ) provided g 0. A real function has the set of real number or one of its subsets both as its domain and as its range. SUPPRTIVE WEB SITES http://www.bbc.co.uk/education/asguru/maths/3pure/0functions/06composite/ inde.shtml http://en.wikipedia.org/wiki/functions http://en.wikipedia.org/wiki/relations MATHEMATICS 49
MDULE - I TERMINAL EXERCISE. Given A a, b, c,, B, 3 Relations -I. Find the number of relations from A to B.. Given that A 7, 8, 9, B 9,0,, C, (i) A B C A B A C (ii) A B C A B A C verify that 3. Which of the following equations represent functions? In each of the case R (a) y 3 4, 4 5 5 (b) 3 y, 0 (c) (d) y, (e) y 4. Write domain and range of the following functions : f : 0,,, 3, 4, 5 6, 7,... 00,0 f :, 4, 4,6, 6, 36... f 3 :,,,,,,,... 3 4 4 f :... 3, 0,,, 4, 5 3 y, 4, 4 6 (f) y 5 f :... 3, 3,,,, 0, 0,,,,... 5. Write domain of the following functions : (a) f 3 (b) f (c) f 3 (d) f 6 (e) f 5 6. Write range of each of the following functions : (b) y, R (a) y 3, R, (c) y, 0,, 3, 5, 7, 9 (All non-negative real values) (d) y, R 50 MATHEMATICS
Relations -I 7. Draw the graph of each of the following functions : (b) y, R (a) y 3, R MDULE - I (c) y, 0,, 3, 5, 7, 9 (d) y, R. 8. Which of the following graphs represent a function? (a) (b) Fig..36 Fig..37 (c) (d) Fig..38 Fig..39 (e) (f) Fig..40 Fig..4 MATHEMATICS 5
MDULE - I (g) (h) Relations -I Fig..4 Fig..43 9. Which of the following functions are rational functions? 3 (a) f, R 4 4 (c) f, R f, R (b) (d) y, R 0. Which of the following functions are polynomial functions? (a) f 3 (b) f (c) f 3 3 4 (d) f 5, 0 (e) f 4,,. Which of the following functions are even or odd functions? (a) f 9 3, 3 (b) f (c) f (d) 5 f (e) (f) Fig..44 Fig..45 5 MATHEMATICS
Relations -I (g) MDULE - I Fig..46. Let f be a function defined by f() = 5 +, R. (i) Find the image of 3 under f. (ii) Find f(3) f() (iii) Find such that f() = 3. Let f() = + and g() = 3 be two real functions. Find the following functions (i) f + g (ii) f g f (iii) f. g (iv) g 4. If f() = ( + 5), g() = are two real valued functions, find the following functions (i) f + g (ii) f g (iii) f g (iv) f g (v) g f MATHEMATICS 53
MDULE - I ANSWERS CHECK YUR PRGRESS..,, 4,,, 4, 4, 4. Relations -I 3. (i) R 4,, 4, 4, 6,, 6, 3, 8,, 8, 4, 0,, 0, 5. (ii) Domain of R 4, 6, 8,0. (iii) Range of R, 3, 4, 5. 4. (i) R, 8,, 4. (ii) Domain of R,. (iii) Range of R, 8, 4 5. (i) R, 4, 3, 9, 5, 5, 7, 49,,, 3,69 Domain of R, 3, 5, 7,,3, Range of R 4, 9, 5, 49,,69 6. (i) Domain of R (ii) Domain of R (iii) Range of R 7. =, y = 3. 8. {(-,-,-), (-,-,), (-,,-), (-,,), (,-,-), (,-,), (,,-), (,,) 9. A={a,b}, B={,y}. 0. 8 CHECK YUR PRGRESS.. (a), (c), (f). (a), (b) 3. (a), (d) 4. (a) Domain, 5, 3, Range,, 5, Range (b) Domain 3,, (c) Domain, 0,,, Range, 0,, (d) Domain Deepak, Sandeep, Rajan, Range 6, 8, 4. 5. (a) Domain,, 3, Range 4, 5, 6 (b) Domain,, 3, Range 4 (c) Domain,, 3, Range,, 3 54 MATHEMATICS
Relations -I (d) Domain Gagan, Ram, Salil, Range 8, 9, 5 (e) Domain a.b, c, d, Range, 4 CHECK YUR PRGRESS.3. (a) (i) Domain = Set of real numbers. (ii) Domain = Set of real numbers. (iii) Domain = Set of a real numbers. (b) (i) Domain R (ii) Domain R, 5 4 3 MDULE - I (iii) Domain R 3, 5 (iv) Domain R 3, 5 (c) (i) Domain R : 6 (ii) Domain R : 7 (iii) Domain 5 : R, (d) (i) Domain : R and 3 5 (ii) Domain : R 3, 5 (iii) Domain : R 3, 7 (iv) Domain : R 3, 7. (a) (i) Range 3, 5, 3, 7, 4 (ii) Range 9, 9, 33, (iii) Range, 4, 8,4, (b) (i) Range f : f (ii) Range 3 f : f 0 (c) (i) Range f : f 5 (ii) Range f : 6 f 6 (iii) Range f : f 5 (iv) Range (d) (i) Range = R (ii) Range = R (iii) Range = R (iv) Range f : f 0 (v) Range f : f 0 f : 0 f 5 (vi) Range f : 0.5 f 0 (vii) Range (viii) Range : All values of f ecept values at 5. MATHEMATICS 55 f :f 0
MDULE - I CHECK YUR PRGRESS.4. (i) No. (ii) Yes Relations -I. (a) (b) Fig..47 Fig..48 (c) (d) Fig..49 Fig..50 (e) (f) Fig..5 Fig..5 3. (c), (d) and (e). CHECK YUR PRGRESS.5. v, vi, vii are true statements. (i), (ii), (iii), (iv) are false statement. 56 MATHEMATICS
Relations -I. (b) (c) are even functions. (d) (e) (h) are odd functions. 3. MDULE - I Fig..53 4. (a) Polynomial function (b) Rational function. (c) Rational function. (d) Rational function. (e) Rational function. (f) Rational function. (g) Constant function. CHECK YUR PRGRESS.6. (i) 4 (ii) 5 (iii) -5. (f+g) =3-, (f-g) =4-, (f.g) = 3, f, g 3 3 TERMINAL EXERCISE. 6 i.e., 64. 3. (a), (b), (c), (d), (e) are functions. 4. f Domain 0,, 4, 6,...00 Range, 3, 5, 7,...0.. Range 4,6, 36,... f Domain, 4, 6,... f3 Domain,,,,... 3 4.. Range,. MATHEMATICS 57
MDULE - I 4. Range 0,, f Domain 3,, 4 5. Relations -I. Range 0,,, 3,... f Domain... 3,,, 0,,, 3,... 5. (a) Domain R. (b) Domain R,. (c) Domain V R. 3. (d) Domain, 6. (e) 5 Domain,. 6. (a) Range R (b) Range All values of y ecept at. 3 4 (c) Range,,,,, 3 3 4 5 (d) Range All values of y for 0 8. (a), (c), (e), (f), (h). Use hint given in check your progress 5.7, Q. No. 7 for the solution. 9. (a), (c) 0. (a), (b), (c). Even functions : (a), (b), (c), (f), (g) dd functions : (d), (e). (i) f(3) = 47 (ii) f(3) f() = 034 (iii) =, 3. (i) f + g = 3 (ii) f g = + 5 (iii) fg = + 6 (iv) f 3, g 3 4. (i) f + g = + + 4 (ii) f g = + + 6 (iii) f. g = 3 + 5 f 5 5 (iv), g (v) g f 5, 5 58 MATHEMATICS