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# Functions (Theory)

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### Functions (Theory)

1. 1. FUNCTIONS
2. 2. Definitions and Shortcuts Relations and Sets • Points to remember 2
3. 3. 1. Let A and B be two non empty sets. The Cartesian product of A and B is A×B which is the set of ordered pairs. A  B   x, y  : x  A, y  B A subset of A×B is called a (binary) relation R from A to B If  x, y   R, then ‘x’ in A is related to ‘y’ in B. We denote it as ‘xRy’. 3
4. 4. 2. The number of elements (members) of set A is denoted by n(A) or O(A) or A . 3. If n(A) is finite then A is called a finite set. Other wise A is an infinite set. 4. If A  m and B  n then A×B = mn and the number of relations from A to B is 2mn . 4
5. 5. 5. Relation on a set A is a relation from A to A which is a R subset R of A  A   x, y  : x,.y  A  A relation R is set to be Reflexive iff xRx  x  A  A relation R is set to be Symmetric iff xRy  yRx  x, y  A  A relation R is set to be Transitive iff  A relation R is set to be Equivalence iff R is Reflexive, Symmetric & Transitive . xRy & yRz  xRz  x, y, z  A. 5
6. 6. 6. If n(A) = m then the number of relations on A is 2 m2 7. If n(A) = m then the number of Reflexive relations on A is 2 m m-1 8. If n(A) = m then the number of Symmetric relations m  m+1 on A is 2 2 9. The number of relations on A which are both reflexive and symmetric is m m-1 2 2 6
7. 7. 10. Partition of a set. An equivalence relation on a set A partitions (divides) it into mutually disjoint subsets such that each member in a subset is related to every member in that subset and not related to members of the other subsets. • If n(A) = m then the number of partitions of A into  ‘r’ disjoint subsets is 1  m  r  r  m m  r -  1   r-1 +  2   r-2  -.... r!       7
8. 8. 11. If n(A) = m then the number of Equivalence relations on A is  r  1 m r m m   r -    r-1 +    r-2  -....  r=1 r!  2  1   m Where ‘r’ is number of disjoint subsets of A. 8
9. 9. 12. If n(A) = m then the number of subsets of A = 2 m. 13. If n(A) = m then the number of proper subsets of A = 2m  1 . 14. The collection of subsets of any given set A is called power set of A and is denoted by P(A). 15 If n(A) = m then the number of elements in P(A) (or) n(P(A))=2m 9
10. 10. 16. Suppose that A consists of the n distinct elements a1,….an, and let 1  r  n . Then number of subsets of A which contain • None of a1,…ar = 2n  r • • Each of a1,…ar = 2n  r At least one of a1,..ar = • Exactly one of a1,..ar = r.2n r • At most one of a1,..ar =  r  1 2n r   2n-r 2r -1 10
11. 11. Functions: • Points to remember 11
12. 12. 1. Definition : Let A and B be two sets and let there exist a rule or manner or correspondence ‘ f ’ which associates to each element of A, a unique element in B. Then f is called a function or mapping from A to B. It is denoted by the symbol f f : A  B or A  B  which reads ‘ f ’ is a function from A to B’ or‘f maps A to B. 12
13. 13. 2. Image and Pre-image: If an element a A is associated with an element b  B then b is called ‘the f image of a’ or ‘image of a under f ’ or ‘the value of the function f at a’. Also a is called the pre-image of b or argument of b under the function f or inverse image of be under f. We write it as b  f  a  or f : a  b or f :  a, b  3. Every function is a relation but every relation need not be a function. 13
14. 14. 4. Domain, Co-domain & Range Of A Function : Let f : A  B , then the set A is known as the domain of f & the set B is known as co-domain of f. The set of all f images of elements of A is known as the range of f . Thus :  Domain of f  a / a   a,f  a    f  Range of f  f  a  / a  A,f  a   B 14
15. 15. 5. Range of a function is always subset to co-domain of the function. 6. The set where the function is well defined is called domain of the function. 7. The set of all images of the elements in the domain of the function is called range of the function. 15
16. 16. 8. If n(A) = m and n(B) = n then the number of functions defined from A to B = nm 9. A function f : A  B is said to be a real variable function iff A  R . 10. A function f : A  B function iff B  R . 11. A function f : A  B is said to be a real function iff A  R , B  R. is said to be a real valued 16
17. 17. Types of functions 1. 2. Polynomial Function : If a function f is defined by f (x) = a0 xn + a1 xn-1 + a2 xn-2 + ... + an-1 x + an where n is a non negative integer and a0, a1, a2, ..., an are real numbers and a 0  0 , then f is called a polynomial function of degree n . Algebraic Function : y is an algebraic function of x, if it is a function that satisfies an algebraic equation of the form n is a P0 (x) yn + P1 (x) yn-1 + ....... + Pn-1 (x) y + Pn (x) = 0 Where positive integer and P0 (x), P1 (x) ...........are Polynomials in x. 3 3 + y3 – 3xy = 0 or y = x 5 e.g. x   17
18. 18. 3. Transcendental Function: A function which is not algebraic is called transcendental function. Ex: y  log x, y  e x etc.. 4. Rational Function: A rational function is a function of the form gx y  f x  h x where g (x) & h (x) are polynomials & h  x   0 . 18
19. 19. 5. Exponential Function : A function f(x) = ax = ex ln a where  a  0,a  0, x  R  is called an exponential function. 6. Logarithmic Function: The inverse of the exponential function is called the logarithmic function . i.e. g(x) = loga x. 7. Absolute Value Function or Mod Function: A function y  f  x   x is called the absolute value function or Modulus function. It is defined as : f x   x    x; if x  0  x 2   0; if x  0  x; if x  0  19
20. 20. 8. Signum Function : A function y= f (x) = Sgn (x) is defined as follows :  1 for x  0  y  f  x    0 for x  0  1 for x  0  or x where x  0  y  f  x   sgn  x    x  0 where x  0  20
21. 21. 9. Greatest Integer Or Step Up Function : The function y = f (x) = [x] is called the greatest integer function where [x] denotes the greatest integer less than or equal to x . if n  x  n  1 where n  I   x   n 10. Fractional Part Function : It is defined as : f (x) = {x} = x - [x]. 21
22. 22. 11. Equal or Identical Function : Two functions f & g are said to be equal if (i)The domain of f = the domain of g. (ii)The range of f = the range of g and (iii)f(x) = g(x) , for every x belonging to their common domain. 1 e.g. f  x   x ,g  x   x2 Where x  0 are identical x functions. 22
23. 23. Fundamental Graphs and Properties of Important Functions
24. 24. Graph of F(x) = 1/x2 24
25. 25. Graph of F(x) = 1/x3 25
26. 26. Comparision of Graphs 1/x, 1/x2, 1/x3, 1/x4 1 x 1 x 2 1 x 2 1 x 3 1 x 3 1 x 26
27. 27. Graph of F(x) = x1/2 27
28. 28. Graph of F(x) = x1/3 28
29. 29. Graph of F(x) = x1/4 29
30. 30. Graph of F(x) = sin x 31
31. 31. Graph of F(x) = cos x 32
32. 32. Graph of F(x) = tan x 33
33. 33. Graph of F(x) = cot x 34
34. 34. Graph of F(x) = sec x 35
35. 35. Graph of F(x) = cosec x 36
36. 36. Graph of F(x) = ax 38
37. 37. Graph of F(x) = ax 39
38. 38. Graph of F(x) = ax 40
39. 39. Graph of F(x) = ax 41
40. 40. Graph of F(x) = loga x 43
41. 41. Graph of F(x) = loga x 44
42. 42. Graph of F(x) = loga x 45
43. 43. Graph of F(x) = loga x 46
44. 44. Graph of Y = |x| 48
45. 45. Properties of Modulus Functions :  x, x  0 (i ) It is defined as y  f ( x)  x  x     x, x  0 (ii) D f  R, R f  [0, ) 2 (iii ) x  a   a  x  a;  a  0  (iv) x  a  x   a and x  a;  a  0  (v) x  y  x  y  x  0 and y  0 or x  0 and y  0 (vi ) x  y  x  y  x  0 and x  y or x  0 and y  0 and x  y (vii ) x  y  x  y (viii ) x  y  x  y 49
46. 46. Graph of F(x)= Sgn  x  x x or ; x0  Definition : F ( x)  Sgn  x    x x  0 ; x0  1; x  0    0; x  0  1; x  0  50
47. 47. Graph of Y = [x] Y 3 2 1 -4 -3 -2 -1 X 1 2 3 4 X -1 -2 -3 -4 Y 51
48. 48. Properties of Greatest Integer Function 1  x  n  n   x  , n  I  2  x   x    x ,  x denotes the fractional part of x.  3   x     x  , x  I  4    x     x   1, x  I  5   x   n  x  n, n  I  6   x   n  x  n  1, n  I 52
49. 49. Properties of Greatest Integer Function  7   x   n  x  n  1, n  I  8  x   n  x  n, n  I  9  n2   x  n1  n2  x  n1  1; n1 , n2  I 10   x  y    x    y  53
50. 50. Properties of Greatest Integer Function  x   x  11      , n  N  n  n  n  1  n  2   n  4   n  8  12         8    16   ....  n, n  N    2   4    1  2 n  1   13  x   x     x    ....   x     nx  , n  N n  n n    54
51. 51. Graph of Y = (x) 55
52. 52. Properties of Least Integer Function 1  x  n   n   x  , n  I  2  x   x    x  1,  x denotes the fractional part of x.  3   x     x  , x  I  4    x     x   1, x  I 56
53. 53. Properties of Least Integer Function  5   x   n  x  n  1, n  I  6   x   n  x  n, n  I  7   x   n  x  n, n  I  8  x   n  x  n  1, n  I  9  n2   x   n1  n2  1  x  n1 ; n1 , n2  I 57
54. 54. Properties of Least Integer Function 10  x  y    x    y   1   x   x  11      , n  N  n  n  n 1  n  2   n  4   n  8  12        ....  2n, n  N  2   4   8   16  1  2 n 1    13  x    x     x    ....   x     nx   n  1, n  N n  n n    58
55. 55. Graph of Y = {x} 59
56. 56. Graph of Y = ax3 + bx2 + cx + d a>0 a<0 60
57. 57. Graph of Y = ax4 + bx3 + cx2 + dx + e a>0 a<0 61
58. 58. Suppose equation is f(x) – g(x) = 0 Or f(x) = g(x) = y (say) then draw the graphs of y = f(x) and y = g(x). If graphs of y = f(x) and y = g(x) cuts at one, two, three,……., no points then no.of solutions are one, two, three,………, zero respectively. Also find f|(x) and g|(x) If f| (x) > g| (x)  y = f(x) is above y = g(x) and If f|(x) < g| (x)  y = f(x) is below y = g(x) 63
59. 59. Example : 1 No.of solutions of the equations y  x   x  and y  1  x 2   Ans : Four Solutions 64
60. 60. Example : 2 No.of solutions of the equations x  sin x Ans : Only One Solution 65
61. 61. Example : 3 No.of solutions of the equations sin x  x 2  x  1 Ans : Zero Solution 66
62. 62. Example : 4 No.of solutions of the equations cos x = x Ans : One Solution 67
63. 63.  1, sin x  0 Graph of y   sin x 1, sin x  0 sin x  1, x   2n ,  2n  1     1, x    2n  1  ,  2n  2   , n  I   69
64. 64. Graph of y = x + sin x Since  1  sin x  1  x  1  x  sin x  x  1 70
65. 65. Graph of y = sin (2x) x Since  1  sin 2  1 71
66. 66. Graph of y = x sin x Since  1  sin x  1   x  x sin x  x 72
67. 67. Graph of y = ex sin x x x Since  1  sin x  1   e  e sin x  e x 73
68. 68. 1. General tips for Sketch The Graphs of Rational Functions :First examine whether denominator has a root or not. If no, then graph is continuous and f is Non-Monotonic. Example. f x  2. x x 2  5x  7 If denominator has roots then f (x) is discontinuous. Such functions can be Monotonic / Non -monotonic. Example: x  x  1 x  2  g  x   f x   x  1 x  2   x  3 x  1  x  1 x  1 h x   x  1 x  2  74
69. 69. 3. If numerator and denominator has a common factor ( say x - a) it would mean removable discontinuity at x = a Example:  x  1 x  1 h x   x  1 x  2  h(x) has removable discontinuity at x = -1 Such a function will always be monotonic i.e. either increasing or decreasing. 75
70. 70. 4. Compute points where the curve crosses the x-axis and also where it cuts the y-axis by putting y = 0 and x = 0 respectively and mark points accordingly. dy 5. Compute dx and find the intervals where f (x) is increasing or decreasing and also where it has horizontal tangent. 6. Find the regions where curve is monotonic. To find whether y is asymptotic or not Compute ‘y’ for x   or x   7. If denominator vanishes say at x = a and (x – a) is not a common factor between numerator and denominator then examine Lim and Lim to find whether f approaches  or   x a x a   76
71. 71. To evaluate the area bounded by the curves, the knowledge of curve tracing is necessary. The following procedure is adopted in order to draw a rough sketch of a function y=f(x) (in cartesian form). 78
72. 72. SYMMETRY i) Symmetry about x-axis : If the equation of the curve involves even and only even powers of y or equation of the curve remains the same by replacing y by –y then the shape of the curve is symmetrical about the x-axis. Y O X 79
73. 73. Example: y2=4ax is symmetrical about x-axis and x2 =4ay is symmetrical about y-axis. Note: The words even and only even should be observed x2+y2 + 2gx + 2fy + c = 0 is not symmetrical about the x-axis. ( Here involves odd power of y as well). 80
74. 74. ii) Symmetry about y-axis: If the equation of the curve involves even and only even powers of x or equation of the curve remains the same by replacing x by –x then the shape of the curve is symmetrical about the y-axis. Y O X 81
75. 75. iii) Symmetry about both axes: If the equation of the curve involves even and only even powers of x as well as of y or equation of the curve remains the same by replacing x by –x & y by –y then the shape of the curve is symmetrical about both the axes. Example : x 2 y2  2 1 2 a b is symmetrical about both axes, Y O X 82
76. 76. iv) Symmetry in opposite quadrants: If the equation of the curve remains unchanged when x and y replaced by –x and –y then the shape of the curve is symmetrical in opposite quadrants. Example: xy=c2 is symmetrical in 1st and 3rd quadrants, as below Y diagram O X 83
77. 77. v) Symmetry about the line y=x: If the equation of the curve remains unchanged when interchanging x and y then the shape of the curve is symmetrical about the line y=x. Example : x3+y3=2axy is symmetrical about the line y=x, as below diagram. Y y=x X 84
78. 78. Origin and Tangents at origin: i) Curve Through Origin: If the point (0,0) satisfies the equation of the curve or the equation of curves does not contain any constant term in addition or subtraction then it passes through the origin. Example : y=x3 passes through the origin. Y O X 85
79. 79. ii) Tangents at the origin: Tangents at the origin are obtained by equating to zero the lowest degree terms occuring in the equation of the curve. Example: The curve x3+y3=3xy passes through the origin and the lowest degree terms occuring in it is i.e., both axes are tangents to the curve at the origin. 86
80. 80. Points of intersection of the curve with the Axes: By putting y=0 in the equation of the curve we get the coordinates of the point of intersection with the x-axis, if they exist. Similarly by putting x=0 in the equation of the curve we get the co-ordinates of the point of intersection with the y-axis, if they exist. Example: Put y=0 in the equation a2x2=y3 (3a-y) then a 2 x 2  0  x  0. Therefore curve meets the x-axis, at (0,0). Put x=0 in the equation then and the curve intersects the y-axis at (0,0) and (0, 3a). 87
81. 81. Regions in which the curve does not lie: If the value of y is imaginary for certain values of x. Similarly if the value of x is imaginary for certain values of y then the curve does not exist for these values of x and y. Example 1: y2=4x For negative value of x we get y has imaginary values . Hence no point of the curve shall exist in the left side of y-axis. Example 2: y2(2a – x)=x3 For x>2a, y is imaginary. There is no curve beyond x=2a ii) For x<0, y is imaginary. Hence no point of the curve shall exist in the left side of y-axis 88
82. 82. Asymptotes Asymptotes are the tangents to the curve at infinity. Working rule for finding the asymptotes of the curve f(x,y) = 0: (Best Method) i) A curve of degree n can not have more than n asymptotes (real or imaginary). ii) Equating to zero the higher degree terms and then factorise. If one factor is y-m1x then corresponding asymptote is y-m1x=c1 where c1 is a constant. y  c1 m1 x  c1 or x i.e., iii) Substituting the value of y i.e., m1 in the equation of curve then equating to zero the higher degree coefficient and find c1. iv) Finally putting the value of c1 in y=m1x+c1, which is the one of the asymptote of the given curve. 89
83. 83. And for the curve y  f ( x) to find asymptotes and kinds of asymptotes remember the following steps: 1. Vertical asymptotes: If at least one of the limits of the function f(x) (at the point a on the right or on the left) is equal to infinity, then the straight line x=a is a vertical asymptote. 90
84. 84. 2. Horizontal asymptotes: If , lim f ( x)  A x   then the straight line y=A is a horizontal asymptote (the right one as x   and the left one as x   ) 3. Inclined asymptotes: If the limits lim f ( x)  k1 , lim [ f ( x)  k1 x]  b1 x x x Exist, then the straight line y  k 1 x  b 2 is an inclined (left) asymptote. A horizontal asymptote may be considered as a particular case of an inclined asymptote at k = 0 91
85. 85. Example 1: Asymptotes of the curve y2 (a2-x2)=x4 Solution. The equation of the curve is x4 + x2y2 – a2y2 = 0 ……(1) Since the curve is of degree 4, therefore it cannot have more than four asymptotes.  x 2  x 2  y 2   0  x 2  x  iy  x  iy   0 Now equating to zero the higher degree terms i.e., x4+x2y2=0 real factor is x2 = 0 or x = 0 92
86. 86. Suppose x =c is an asympote then put x=c in …..(1) c4+c2y2 – a2y2 = 0 Equating the higher degree coefficient = 0 Then c2 – a2 = 0 or c   a Then asymptotes are x  a which are parallel to y-axis. 93
87. 87. Example 2. Asymptotes of the curve y2(x2 – a2)= x2(x2 – 4a2). Solution. The equation of the curve is y2x2 - a2y2 –x4+4a2 x2 = 0 or y2x2 – x4 +4a2x2 – a2y2 = 0 ……….(1) Since the equation of the curve is of degree 4, therefore it can not have more than four asymptotes. Equating to zero the higher degree terms i.e., y2x2 – x4 = 0  x 2  y2  x 2   0  x 2  y  x  y  x   0 Real factors are x = 0, y = -x, y = x Suppose x=c1, y= - x+c2, y = x + c3 are the asymptotes 94
88. 88. Putting x=c1 in (1) then c12 y 2  c14  4a 2 c12  a 2 y 2  0 Equating the higher degree coefficient = 0 Then c12  a 2  0 or c1  a Then asymptote are x  a Again putting y=-x+c2 in (1) then 2 2    x  c 2  x 2  x 4  4a 2 x 2  a 2   x  c 2   0  x 4  2x 3c 2  c 2 x 2  x 4  4a 2 x 2  a 2 x 2  a 2 c2  2a 2 c 2 x  0 2 2  2x 3c 2  x 2  c 2  3a 2   2a 2 c 2 x  a 2 c 2  0 2 2 Equating higher degree coefficient = 0  c2  0 95
89. 89. Then asymptote is x+y = 0 In last putting y = x+c3 in (1) then  x  c3  2 2 4 2 2 2 2 x  x  4a x  a  x  c3   0 2 2  x 4  2x 3c3  c3 x 2  x 4  4a 2 x 2  a 2 x 2  a 2 c3  2a 2 c3 x  0 2 2  2x 3c3  x 2  c3  3a 2   2a 2 c3 x  a 2 c3  0 Equating the higher degree coefficient = 0 or c3 = 0 then asymptote is x – y = 0 Finally, all asymptotes are x  a, y   x . 96
90. 90. Tangent Put dy  0 for the points where tangent is parallel to the x-axis and put dx dx 0 dy for the points where the tangent is parallel to y-axis. 97
91. 91. Points of Maxima and Minima: First find the critical point i.e., then minima and if d2 y 0 2 dx dy  0 or dx does not exist. If d2 y 0 2 dx then maxima at that point. For maxima & minima odd derivative must be = 0, if even derivative +ve then minima and if even derivative – ve then maxima at that point. 98
92. 92. Concavity and Points of Inflection: a) Concave up The graph of a differentiable function y=f(x) is concave up on an interval if increases or the graph y=f(x) is concave up on any interval if d2 y  f ''  x   0 2 dx O Note: For concave up slope increase from positive direction of axis or from negative direction of axis according as value of x increases or decreases. 99
93. 93. b) Concave down: The graph of a differentiable function y=f(x) is concave down on an dy interval if decreases or the graph y= f(x) is concave down on any dx interval if d2 y  f 11  x   0 dx 2 O Note: For concave down slopes decreases from positive direction of x-axis or from negative direction of x-axis according as value of x increases or decreases. 100
94. 94. c) Inflection: A point on a curve y=f(x) if the concavity changes from up to down or d2 y  0 at 2 dx down to up is called a point of inflection and if a point that is not a point of inflection. Example . The curve y=x3 has a point of inflection at x=0 Y 2 d y  6x 2 dx Where . Solution. Since and y = x3 2 d y  0 at x  0 dx 2 d2 y  0 at x  0 2 dx i.e., sign changes of X' O X d2 y at x  0 2 dx Y' 101
95. 95. Node and Cusp: A double point is called node at which two real tangents (not coincident) can be drawn and a double point is called cusp at which two tangents at it are coincident. Y Y . . Node Cusp A O A X O X 102
96. 96. Table Prepare a table for certain values of x and y. Example: y  x x 0 1 2 3 4 5 y 0 1 2 3 2 5 6 6 Note: 1. Taking at least four values of x. 2. Taking scale on both axes is same 103
97. 97. One-One and Many-One Functions If each element in the domain of a function has a distinct image in the co-domain, the function is said to be One-One. One-one functions are also called injective functions. On the other hand, if there are at least two elements in the domain whose images are the same, the function is known as Many-one. Note: 1. A function will be either one-one or many one. 2. A many-one function can be made one-one by redefining the domain of the original function. 105
98. 98. Methods to Determine One-One and Many-One Graphical Lines drawn parallel to the x-axis from the each corresponding image point should intersect the graph of y=f(x) at one (and only one) point if f(x) is one-one and there will be at least one line parallel to x-axis that will cut the graph at least at two different points if f(x) is many-one and vice versa. 106
99. 99. y f(x) = 2x + 5 x 0 Graph of f(x) = 2x + 5 107
100. 100. f  x  x 2  1 y  x2  x1 0 x1 x2 x Graph of f  x   x 2  1 108
101. 101. Analytical Method: a. Let x1 , x2  domain of f and if x1  x2  f  x1   f  x2  for every x1 , x2 in the domain, then f is one-one else many-one. b. Conversely, if f  x1   f  x2   x1  x2 for every x1 , x2 in the domain, then f is one-one else many-one. 109
102. 102. Calculus Method: c. If the function is entirely increasing or decreasing in the domain, then f is one-one else many-one. d. Any continuous function f(x) that has at least one local maxima or local minima is many-one. 110
103. 103. e. All even functions are many-one. f. All polynomials of even degree defined in R have at least one local maxima or minima and hence are many-one in the domain R. Polynomials of odd degree can be one-one or many-one. 111
104. 104. g. If f is a rational function, then f  x1   f  x2  will always be satisfied when x1  x2 in the domain. Hence, we can write f  x1   f  x2    x1  x2  g  x1 ,x2  where g  x1 ,x2  is some function x1 and x2 . Now, if g  x1 ,x2   0 gives some solution which is different from x1  x2 and lies in the domain, then f is many-one else one-one. 112
105. 105. Onto and Into Functions Let f : X  Y be a function. If each element in the co-domain Y has at least one pre-image in the domain X, that is, for every y  Y there exists at least one element x  X such that f(x)=y , then f is onto. In other words, the range of f = Y for onto functions. On the other hand, if there exists at least one element in the codomain Y which is not an image of any element in the domain X, then f is into. i.e., A function which is not onto then it is an into Onto function is also called surjective function. 113
106. 106. Methods to Determine Onto or Into Analytical : a. If range = co-domain, then f is onto. If range is a proper subset of co-domain, then f is into. b. Solve f(x)=y for x, say x = g(y). Now if g(y) is defined for each y  co-domain and g  y   domain of f for all y  co-domain, then f(x) is onto. If this requirement is not met by at least one value of y in the codomain, then f(x) is into. 114
107. 107. Note: a. An into function can be made onto by redefining the codomain as the range of the original function. b. Any polynomial function f :R R is onto if degree is odd; into if degree of f is even. 115
108. 108. One-One, Onto Function Or Bijection If a function f : X  Y is both one-one and onto then it is called a bijective function. Note: 1. A function f : X  Y is one-one only if n(X)  n(Y) 2. A function f : X  Y is onto only if n(X)  n(Y) 3. A function f : X  Y is a bijection only if n(X)  n(Y) 4. If n(X) = n(Y)= n, then no.of one-one functions defined from X to Y = no.of onto functions defined from X to Y = no.of bijections defined from X to Y = n! 116
109. 109. Number of Functions (Mappings) Consider set A has n different elements and set B has r different elements and function f : A  B Description Equivalent to Number of functions number of ways in which n different balls can be distributed among r persons if 1. Total number of functions Any one can get any number of objects rn 117
110. 110. Description Equivalent to number Number of functions of ways in which n different balls can be distributed among r persons if 2. Total number of one-to-one function Each gets exactly 1 objects or permutation of n different objects taken r at a time  r Cn  n !, r  n  rn  0, 118
111. 111. Description Equivalent to Number of functions number of ways in which n different balls can be distributed among r persons if 3. Total At least one gets number of more than one ball many-One functions r n  n Cn . n !, r  n  n rn r , 119
112. 112. Description Equivalent to Number of functions number of ways in which n different balls can be distributed among r persons if 4. Total number of onto functions Each gets at least one ball rn r C  r 1n r C2  r 2n r C3  r 3n ...., r n 1  r !, r n   0, r n  120
113. 113. Description Equivalent to Number of functions number of ways in which n different balls can be distributed among r persons if 5. Total number of into Function Which is not onto rC  r1n r C  r2n r C  r 3n ...., r n  1 2 3  rn, r n   121
114. 114. Description Equivalent to number of ways in which n different balls can be distributed among r persons if Number of functions 6. Total number of Constant Functions All the balls are received by any one person r 122
115. 115. Identity Function A function f : X  X is said to be an identity function. If f (x)  x, x  X Note : 1. Every identity function is bijective function. 2. If n(X)=n, then no.of identity functions defined on X = 1 3. Usually identity function is also defined on a set A is denoted by I or IA. 4. Every identity function is a bijective but converse need not be true 123
116. 116. Constant Function A function f : A  B is a constant. If there exists k  B such that f (a)  k, a  A . Note : 1. Every constant function is a many one function but converse need not be true 2. If n(A) = n and n(B)=m then no.of constant mappings defined from A to B= m 3. If range of any function is a singleton set then the function is a constant 124
117. 117. Composite Function Let A, B and C be three non-empty sets. Let f : A  B and g : B  C be two functions then gof : A  C. This function is called the product or composite of f and g, given by (gof )x  g{f (x)}x  A 125
118. 118. A f B x y=f(x) g C z = g {f(x)} gof : A  C Thus the image of every x  A under the function gof is the g-image of the f-image of x. 126
119. 119. Note: 1. The gof is defined only if x  A,f (x) is an element of the domain of g so that we can take its g-image 2. The range of f must be a subset of the domain of g in gof 3. (i) (fog)x=f{g(x)} (ii) (fof)x=f{f(x)} (iii) (gog)x=g{g(x)} (iv) (fg)x=f(x).g(x) (v) (f  g)x  f (x)  g(x) f  f (x) x ;g(x)  0 g (vi)   g(x) 127
120. 120. Properties of Composite Functions a. The Composition of functions is not commutative in general, i.e., fog  gof b. The Composition of functions is associative i.e., if h : A  B, g : B  C and f : C  D be three functions, then (fog)oh = fo(goh) 128
121. 121. c. The composition of any function with the identity function is the function itself, i.e., f : A  B then foI A  I B of  f where IA and IB are the identity functions of A and B, respectively. d. If f : A  B and g : B  C are one-one, then gof : A  C is also one-one. 129
122. 122. e. If f : A  B and g : B  C are onto, then gof : A  C is also onto. f. If gof(x) is one-one, then f(x) is necessarily one-one but g(x) may not be one-one. Consider the function f(x) and g(x) as shown in the following figure. 130
123. 123. f g B B C 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 6 5 5 A 6 (b ) (a) Here f is one-one, but g is many-one. But g(f(x)): {(1,1), (2,2), (3,3), (4,4)} is one-one. 131
124. 124. g. If gof(x) is onto, then g(x) is necessarily onto but f(x) may not be onto. g f A B B C 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 (a) (b) 132
125. 125. Here, f is into and g is onto. But (gof)(x): {(1,1), (2,2), (3,3), (4,3)} is onto. Thus, it can be verified in general that gof is one-one implies that f is one-one. Similarly, gof is onto implies that g is onto. 133
126. 126. Inverse Function If f : A  B is a bijection then f 1 : B  A is called inverse of f and is defined as for a  A then  a unique b  B s.t f (a)  b  f 1 (b)  a f B A a b = f( a ) f -1 134
127. 127. The Graph of the Inverse Function In considering the inverse (if any) of the real-valued function y = f(x) of a real variable, this function is regarded as a function from its domain onto its range; it is therefore invertible if and only if it is one-one. Suppose that the function y= f(x) is invertible. We describe the relationship between the graph S of y = f(x) and the graph S of y  f 1 (x) 135
128. 128. Inverse Function y  f 1 (x) y P(a,b) y = f(x) Q(b,a) O y= x x 136
129. 129. The point P=P(a, b) lies on S if and only if b  f 1 (a) , or equivalently a = f(b), which means that the point Q= Q(b, a) lies on S. Since the points P, Q are reflections of each other in the line y = x [because this line bisects the segment PQ at right angles], it follow that: The graph S of y  f 1 (x) is the reflection in the line y =x of the graphs S of y =f(x) 137
130. 130. Properties of Inverse Function • The inverse of bijective function is unique and bijective • Let f : A  B be a function such that f is bijective and g : B  A is inverse of f, then fog = IB= identity function of set B. Then gof = IA = identity function of set A. • If fog=gof then either f 1  g or g 1  f and fog(x)=gof(x)=x 138
131. 131. • If f and g are two bijective functions such that f : A  B and g : B  C ,then gof : A  C is bijective. Also (gof ) 1  f 1og 1 • Graphs of y = f(x) and y  f 1 (x)are symmetrical about the line y = x and intersect on the line y = x or f (x)  f 1 (x)  x whenever graphs intersect. 139
132. 132. y y y  f 1 (x) y = f(x) x y = (x) O (-1,0) y = f(x) x (0,-1) y (1) y  f 1 (x) x O (2) 140
133. 133.  x  4, x  [1, 2] But in the case of the function f (x)     x  7, x  [5, 6]  x  4, x  [5,6] f (x)   7  x, x  [1, 2] 1 y = f(x) and y  f 1 (x) intersect at (3/2, 11/2) and (11/2, 3/2) which do not lie on the line y =x 141
134. 134. y y=x 6 5 4 3 2 1 0 x 1 2 3 4 5 y=x 6 y  f 1 (x) 142
135. 135. EVEN AND ODD FUNCTIONS Even Function A function y = f(x) is said to be an even function if f   x   f  x   x  D f . Graph of an even function y = f(x) is symmetrical about the y-axis, i.e., if point (x, y) lies on the graph then (-x, y) also lies on the graph. 143
136. 136. y y  x2 x' x O  a Y yx y x X' 4  5 4  5 X Y'  b 144
137. 137. Odd Function A function y = f(x) is said to be an odd function if f   x    f  x   x  D f . Graph of an odd function y = f(x) is symmetrical in opposite quadrants, i.e., if point (x, y) lies on the graph then (-x, -y) also lies on the graph 145
138. 138. y y yx y  x3 x' x O (1) x O (2) y y  sin x x O (3) 146
139. 139. Properties of odd and Even Functions • Sometimes, it is easy to prove that f(x)-f(-x)=0 for even functions and f(x)+f(-x)=0 for odd functions. • A function can be either even or odd or neither. 147
140. 140. • Every function (not necessarily even or odd) can be expressed as a sum of an even and an odd function, i.e.,  f  x  f  x   f  x  f  x   f  x     2 2     Let  f  x  f  x   f  x  f x  h  x    and g  x     2 2     It can now easily be shown that h(x) is even and g(x) is odd. 148
141. 141. • The first derivative of an even function is an odd function and vice versa. • If x  0  domain of f, then for odd function f(x) which is continuous at x=0, f(0)=0, i.e., if for function, f (0)  0 , then that function cannot be odd. It follows that for a differentiable even function f '  0   0, i.e., if for a differentiable function f '  0  0 then the function f cannot be even. 149
142. 142. • f(x)=0 is the only function which is defined on the entire number line is even and odd at the same time. • Every even function y=f(x) is many-one . x  D f 150
143. 143. f  x g  x f  x  g  x f  x  g  x f  x g  x f  x / g  x fog  x  Even Even Even Even Even Even Even Even Odd Odd Even Even Neither even nor odd Neither even nor odd Odd Odd Neither even nor odd Neither even nor odd Odd Odd Even Odd Odd Odd Odd Even Even Odd 151
144. 144. Periodic Functions A function f : X  Y is said to be periodic function if there exists a positive real number T such that f  x  T   f  x  , x  X The least of all such positive numbers T is called the principle period or fundamental period of f. All periodic functions can be analyzed over an interval of one period within the domain as the same pattern shall be repetitive over the entire domain. 152
145. 145. Properties of Periodic Functions • If f(x) is periodic with period T, then af  x  b   c where a, b, c  R  a  0  is also periodic with period T. • If f(x) is periodic with period T, then f(ax+b) where T. a, b  R  a  0  is also period with period a 153
146. 146. • m Let f(x) has period p  m, n  N and co-prime and n r g(x) has period q  , r , s  N ( and co-prime) then s LCM of m, r  period of f+g= LCM of p and q, i.e., t  . HCF of n, s  t will be the period of (f+ g)provided there does not exist a positive number k(<t) for which f  xk g xk  f  x g x, else k will be the period. 154
147. 147. • The same rule is applicable for any other algebraic combination of f(x) and g(x). • LCM of p and q exists if p and q are rational quantities. If p and q are irrational, then LCM of p and q does not exist unless they have same irrational surd. LCM of rational and irrational is not possible. 155
148. 148. • sin n x,cos n x,cos ec n x and secn x have period 2 if n is odd and  if n is even. • tan n x and cot n x have period  whether n is odd or even. • A constant function is periodic but does not have a Fundamental period. • If g is periodic, then fog will always be a periodic function. Period of fog may or may not be the period of g 156
149. 149. • If f is periodic and g is strictly monotonic (other than linear) then fog is non-periodic. • A continuous periodic function is bounded. • If f(x), g(x) are periodic functions with periods T1, T2, respectively, then, we have h(x) = f(x) + g(x) has period as 157
150. 150. a. LCM of {T1, T2}; if f(x) and g(x) cannot be interchanged by adding a least positive number less than the LCM of {T1, T2}. b. k; if f(x) and g(x) can be interchanged by adding a least positive number k(k< LCM of {T1, T2}). 158
151. 151. Example:- Consider the function f  x  sin x  cos x , |sinx| + |cosx| have period , hence according to the rule of LCM, period of f(x) is  .        x     sin  x     cos  x    But f                2 2 2   cos x  sin x . Hence, period of f(x) is . 2 159
152. 152. DOMAIN AND RANGE
153. 153. Domain & Algebra of Domain Let f : A  B is a function from A to B, then the set A is called the domain of the function f (denoted by Df) and the set B is called the Co-domain of the function f (denoted by Cf). The set of all those elements of B which are the images of the elements of set A is called the range of the function f (denoted by Rf). Domain Of f  Df  {a : a  A,(a,f (a))  f} Range of f  R f  {f (a) : a  A,f (a)  B} 161
154. 154. Algebra of the domain of the Function: • Domain of (f (x)  g(x)) = Domain of f (x)  Domain of g(x) i.e., Df g  Df  Dg • Domain of (f(x).g(x)) = Domain of f (x)  Domain of g(x) i.e., Dfg  Df  Dg  f (x)     • Domain of  g(x)  = Domain of f (x)  Domain of     g(x)  {x : g(x)  0} i.e., Df /g  Df  Dg  {x : g(x)  0} 162
155. 155. • Domain of f (x) = Domain of f (x)  {x : f (x)  0} i.e., D • f  Df  {x : f  0} Domain of log a f (x) = Domain of f (x)  {x : f (x)  0} i.e., Dloga f  Df  {x : f  0} • Domain of (fog)x= Domain of g(x) i.e., Dfog =Dg [Where (fog)x=f{g(x)}] 163
156. 156. How to find Range of a Function? Let f(x) be any given real function Step-1 Find the Df Step-2 • If Df is finite set, then find images of every element in Df then the set of collection of all images of the elements in Df is the range of the function 164
157. 157. • If Df  R (which is not an interval) then consider f(x) as y, and find the x in terms of y. Then the collection of all the values of y where x is real is nothing but the range of the function. • Of Df is an interval (closed/open/semi closed/semi open) then test the monotonicity of f in Df and find its least and greatest values. Then range of the function becomes Least value of f  y  greatest valueof f 165
158. 158. Remember • A function f is said to be increasing if f (x)  0 x  D f • A function f is said to be decreasing if f (x)  0 x  Df • If f is increasing in [a, b] then range f = [f(a), f(b)] • If f is decreasing in [a, b] then range f = [f(b), f(a)] 166
159. 159. The Greatest and Least Values of a Continuous Function Let y =f(x) be a given function in an interval [a,b]. The greatest and least values of a continuous function f(x) in an interval [a,b] are attained either at the critical points of f(x) within [a,b] or at the end points of the interval. 167
160. 160. i) The Greatest/Largest values of a function in interval [a,b]: Find out the critical point of f(x) in (a, b). Let 1 , 2 , 3 ,......, n be the critical points and also find the values of the function at these critical points i.e., f (1 ),f (2 ),f (3 ),......,f (n ) be the values of the function at critical points. Then the greatest value of the function f(x) in [a, b] is given by G  max{f (a),f (1 ),f (2 ),f (3 ),...f (n ),f (b)} and least value of the function f(x) in [a,b] is given by L  min {f (a),f (1 ),f (2 ),f (3 ),....f (n ),f (b)} 168
161. 161. ii) The Greatest/Largest values of a function in interval (a,b): Find out the critical points of f(x) in (a, b). Let 1 , 2 , 3 ,......, n be the critical points and also find the values of the function at these critical points i.e., f (1 ),f (2 ),f (3 ),......,f (n ) be the values of the function at critical points. Then the greatest value of the function f(x) in (a, b) is given by G  max{f (1 ),f (2 ),f (3 ),...f (n )} and least value of the function f(x) in (a, b) is given by L  min{f (1 ),f (2 ),f (3 ),...f (n )} 169
162. 162. Note: 1) If xlim f (x) and xlim f (x)  G or < L then f(x) would not a  b have Greatest or Least value of (a, b) 2) If f (x)   as x  a or x  b and f (x)  0 only for one value of x (say c) between a and b, then f(c) necessarily minimum and the global minimum. 170
163. 163. and if f (x)   as x  a or x  b and f (x)  0 . Only for one value of x (say c) between a and b, f(x) is necessarily maximum and the global maximum. 3) If f(x) is a continuous function in its domain then between two maxima there is one minimum and between two minima there is one maximum 171
164. 164. Domain & Range of Standard Functions S.NO FUNCTION DOMAIN RANGE 1 log a x  a  1, a  0  R   (0, ) 2 a x (a  0) R 3 [x] R Z 4 [ax+b] R Z R  ( ,  ) R   (0, ) 172
165. 165. S.NO 5 FUNCTION {x}=x-[x] DOMAIN RANGE R [0.1) 6 {ax+b} R  b 1  b   a , a   7 |x| R 0,   R  b    a ,    8 |ax+b| 173
166. 166. S.NO 9 10 FUNCTION DOMAIN RANGE x  0,    0,   ax  b  b    a ,    0,   11 sinx R [-1,1] 12 cosx R [-1,1] 174
167. 167. S.NO FUNCTION DOMAIN RANGE 13 sin(ax+b) R [-1,1] 14 cosx R [-1,1] tanx    R   2n  1 / n  Z  2   R tan(ax+b)  b   R   2n  1  / n  Z  R 2a a   15 16 175
168. 168. S.NO FUNCTION 17 18 19 20 DOMAIN RANGE cotx R  n / n  Z R cot(ax+b)  n b  R    / n  Z a a  R secx    R   2n  1 / n  Z  2    , 1  1,   sec(ax+b)  b    , 1  b   1  b ,   R   2n  1  / n  Z      a   a  2a a    176
169. 169. S.NO 21 22 FUNCTION DOMAIN RANGE  , 1  1,   cosecx R  n / n  Z cosec(ax+b)  n b  R    / n  Z a a  1  b   1  b    ,     a , a    [-1,1]     2 , 2    [-1,1] 0,  1 23 s in 24 cos 1 x x 177
170. 170. S.NO 25 FUNCTION tan x 26 cot 1 x 27 sec RANGE R 1 1 DOMAIN x 1 28 cos ec x     ,   2 2 R  0,   , 1  1,    0,     2  , 1  1,        2 , 2   0   178
171. 171. S.NO FUNCTION DOMAIN RANGE 29 sin hx R R 30 cos hx R 1,   31 tan hx R (-1,1) 32 cot hx  , 0   0,    , 1  1,   179
172. 172. S.NO FUNCTION DOMAIN RANGE 33 sec hx R (0,1] 34 cosec hx  , 0    0,    ,0    0,   35 sinh 1 x R R 1,    0,   cosh 1 x 36 180
173. 173. S.NO FUNCTION 37 tanh 1 x DOMAIN RANGE (-1,1) R  , 1  1,    , 0    0,   sec h 1x 39 (0,1] 0,   cos ech 1x 40  , 0    0,    , 0    0,    , 0    0,    , 2   2,   38 41 coth 1 x 1 x x 181
174. 174. Standard Results •  a 2  b 2 , a 2  b 2  Range of asinx + bcosx is   • Range of asinx + bcosx +c is  c  a 2  b 2 ,c  a 2  b 2    • Range of asinx + b is [|b-|a|, b+|a|] • Range of acosx + b is [|b-|a|, b+|a|] • Range of f (x)  cos x sin x  (sin 2 x  sin 2    is  (1  sin 2 , 1  sin 2     182
175. 175. • Range of f (x)  (a 2 cos 2 x  b 2 sin 2 x  (a 2 sin 2 x  b 2 cos 2 x), a  b is a  b, 2(a 2  b2 )    • If ,   are three real numbers,  positive and  non-zero x 2   x   then the range of the function f (x)     x   x 2 is R if and only if    and           183
176. 176. • (a  x)(b  x) Minimum value of is (c  x)  a c  bc  2 where a > c, b > c and for every x > -c • Minimum value of 2(a  x)  (x  x 2  b 2 ) is a 2  b 2 where x  R 184
177. 177. STANDARD LOGARITHEMIC INEQUALITIES:
178. 178. Sign Properties 1. log a x  0  x  1, a  1 or 0  x  1, 0  a  1 2. log a x  0  x  1, 0  a  1 or 0  x  1, a  1 186
179. 179. Inequalities I (i) If a  1, then x  y  log a x  log a y (ii) If 0  a  1, then x  y  log a x  log a y II (i) If a  1, then x  a  log a x  1 (ii) If a  1, then x  a  0  log a x  1 (iii) If 0  a  1, then x  a  log a x 1 (iv) If 0  a  1, then x  a  0  log a x  1 187
180. 180. III (i) a  1, x  1  0  log a x  0 (ii) 0  a  1, x  1 log a x  0 (iii) 0  a  1, 0  x  1  log a x  0 (iv) a  1, 0  x  1  log a x  0 188
181. 181. IV (i) if a  1, and log a x  m, then x  a m (ii) if a  1, and log a x  m, then x  a m (iii) if 0  a  1, and log a x  m, then x  a m (iv) if 0  a  1, and log a x  m, then x  a m 189
182. 182. Some More Standard Inequalities 1. a  b and b  c  a  c 2. a  b  a  c  b  c and a  c  b  c c 3. a b a  b and c  0  ac  bc and  c c 4. a b a  b and c  0  ac  bc and  c c 5. a  b and n  0  a n  b n , a1/ n  b1/ n and a  n  b  n 6. A.M  G.M  H.M 190
183. 183. 7. Theorem of Weighted Means: Let a1 , a 2 ,....., a n be positive real numbers and m1 , m2 ,....., mn be n positive rational numbers. Then:  m1a1  m 2a 2  ...  m n a n   m1  m 2  ...  m n  m m m  a1 1 .a 2 2 ...a n n     1  m1  m2 .... mn  191
184. 184. 8. Cauchy-Schwartz Inequality: If a1 , a 2 ,....., a n and b1 , b2 ,....., b n are any two sets of real numbers, then 2  2 2  a1b1  a 2 b 2  ......  a n b n   a1  a 2  .....  a n 2  2 2 b1  b2  .....  b n 2  a1 a 2 an and equality holds when b  b  ........  b 1 2 n 192
185. 185. 9. Weierstrass Inequality: (i) Let a1 , a 2 ,....., a n be positive real numbers, then (1  a1 )(1  a 2 ).....(1  a n )  1  a1  a 2  .....  a n (ii) Let a1 ,a 2 ,.....,a n be positive real numbers, each less than 1, then (1  a1 )(1  a 2 ).....(1  a n )  1  a1  a 2  .....  a n 193
186. 186. 10. Tchebychef’s Inequality: If a 1 , a 2 , ....., a n and b1 , b2 ,....., bn are any two sets of real numbers, such that a1  a 2  .......  a n and b1  b2  .......  bn then: (i) n  a1b1  a 2 b2  ......  a n bn    a1  a 2  .....  a n  b1  b2  .....  b n  a b  a b  ......  a n b n   a1  a 2  ......  a n   b1  b 2  ......  b n  (ii)  1 1 2 2  . n n n 194
187. 187. 11. If a and b are distinct positive real numbers and m is rational number different from 0 and 1, then m  a m  bm   a  b  (i)    : if 0  m  1 2   2   m  a m  bm   a  b  (ii)  2    2  : if m  0 or m  1     195
188. 188. 12. Let a1 , a 2 ,.....,a n be positive real numbers, and m is a positive rational numbers different from 0 and 1, then: (i) a m 1 m  a 2  ......  a n m n (ii)  m m a1  a 2  ......  a n n  a m 1  a 2  ......  a n    n    a   m if 0  m  1 m 1  a 2  ......  a n   if m  0 or m  1 n  196
189. 189. Sign Scheme of Trigonometric Functions Inequality Sinx > k = sin  Sol in  0, 2 Or  ,  General Solution x   ,     x   2n  , 2n      x   0,       , 2  x   2n, 2n      2n    , 2n  2  x   ,   x   2n  , 2n    Sinx < k = sin  Cosx > k = cos  197
190. 190. Inequality cosx < k = cos  tanx > k = tan  tanx < k = tan  Sol in  0, 2 Or  ,  x  (, 2  ) General Solution x  (2n  , 2n  2  )  3      x   ,      ,  x   n  , n   2 2    2       x  ,    , 2   2     x   n  , n    2   198