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. 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. 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. 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. 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. 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. 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. 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
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. 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. 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. 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. 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. 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. 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
gx
y f x
h x
where g (x) & h (x) are polynomials & h x 0 .
18
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. 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. 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. 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
49. 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
50. Graph of F(x)= Sgn x
x
x
or
; x0
Definition : F ( x) Sgn x x
x
0
; x0
1; x 0
0; x 0
1; x 0
50
51. Graph of Y = [x]
Y
3
2
1
-4
-3
-2
-1
X
1
2
3
4
X
-1
-2
-3
-4
Y
51
52. 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
53. 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
54. 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
56. 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
57. 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
58. 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
61. Graph of Y = ax4 + bx3 + cx2 + dx + e
a>0
a<0
61
62.
63. 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
64. Example : 1
No.of solutions of the equations
y x x and y 1 x 2
Ans : Four Solutions
64
65. Example : 2
No.of solutions of the equations
x sin x
Ans : Only One Solution
65
66. Example : 3
No.of solutions of the equations
sin x x 2 x 1
Ans : Zero Solution
66
67. Example : 4
No.of solutions of the equations cos x = x
Ans : One Solution
67
68.
69. 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
70. Graph of y = x + sin x
Since 1 sin x 1
x 1 x sin x x 1
70
71. Graph of y = sin (2x)
x
Since 1 sin 2 1
71
72. Graph of y = x sin x
Since 1 sin x 1 x x sin x x
72
73. Graph of y = ex sin x
x
x
Since 1 sin x 1 e e sin x e
x
73
74. 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
75. 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
76. 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
77.
78. 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
79. 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
80. 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
81. 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
82. 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
83. 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
84. 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
85. 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
86. 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
87. 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
88. 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
89. 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
90. 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
91. 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
92. 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
93. 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
94. 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
95. 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
96. 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
97. 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
98. 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
99. 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
100. 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
101. 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
102. 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
103. 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
104.
105. 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
106. 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
107. y
f(x) = 2x + 5
x
0
Graph of f(x) = 2x + 5
107
108. f x x 2 1
y
x2 x1
0
x1
x2
x
Graph of f x x 2 1
108
109. 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
110. 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
111. 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
112. 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
113. 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
114. 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
115. 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
116. 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
117. 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
118. 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
rn
0,
118
119. 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
rn
r ,
119
120. 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 1n r C2 r 2n r C3 r 3n ...., r n
1
r !,
r n
0,
r n
120
121. 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 r1n r C r2n r C r 3n ...., r n
1
2
3
rn,
r n
121
122. 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
123. 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
124. 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
125. 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
126. 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
127. 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
128. 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
129. 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
130. 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
132. 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
133. 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
134. 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
135. 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
137. 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
138. 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
139. •
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
140. 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
141. 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
143. 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
145. 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
147. 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
148. •
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
149. •
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
150. •
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
151. 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
152. 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
153. 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
154. •
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 xk g xk f x g x,
else k will be the period.
154
155. •
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
156. •
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
157. •
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
158. 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
159. 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
161. 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
162. 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
163. •
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
164. 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
165. •
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
166. 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
167. 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
168. 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
169. 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
170. 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
171. 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
172. 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
176. 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
177. 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
178. 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
182. 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
183. •
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
184. •
(a x)(b x)
Minimum value of
is
(c x)
a c bc
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
186. 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
187. 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
188. 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
189. 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
190. 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
191. 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
192. 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
193. 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
194. 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
195. 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
196. 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
197. 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
198. 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
