Hg
Name: Class 11 Date:
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
Relations and Functions
• Ordered pair: A pair of elements grouped together in a particular order.
• Cartesian product: A × B of two sets A and B is given by
A × B = {(a, b): a Є A, b Є B}
In particular R × R = {(x, y): x, y Є R} and R × R × R = (x, y, z): x, y, z Є R}
• If (a, b) = (x, y), then a = x and b = y.
• If n(A) = p and n(B) = q, then n(A × B) = pq.
• A × Ф = Ф
• In general, A × B ≠ B × A.
• Relation: A relation R from a set A to a set B is a subset of the cartesian product A × B obtained by describing a
relationship between the first element x and the second element y of the ordered pairs in A × B.
• The image of an element x under a relation R is given by y, where (x, y) Є R,
• The domain of R is the set of all first elements of the ordered pairs in a relation R.
• The range of the relation R is the set of all second elements of the ordered pairs in a relation R.
• Types of Relations:
o Empty Relation: A relation R from set X to X is an empty relation given by R = Ф X x X.
o Universal Relation: A relation from set X to X is universal relation given by R = X x X.
o Identity Relation: A relation from set X to X is an Identity relation if (a, a) Є R only for every a Є X.
o Reflexive Relation: A relation R from set X to X is said to be reflexive if (a, a) Є R, for every a Є X.
e.g. "is equal to", "divides" are reflexive while "is not equal to", “is greater than” are irreflexive.
o Symmetric Relation: A relation R from set X to X is symmetric if (a, b) Є R (b, a) Є R, for every a, b Є X.
e.g. "is married to”, “is a team mate of”, “spouse of”, “sibling of” are symmetric. “is greater than”, “is less
than” are asymmetric.
o Transitive Relation: A relation R from set X to X is said to be transitive if (a, b) Є R and (b, c) Є R (a, c) Є R,
for every a, b, c Є X.
e.g. “sibling of”, "is a subset of", “divides”, “is greater than” are transitive. “is perpendicular to”, “is an
immediate successor of” are non-transitive.
o Equivalence Relation: A relation which is reflexive, symmetric and transitive is called an equivalence relation.
e.g. “is equal to”, “is similar to”, “is congruent to”, “has the same birthday as”.
o Equivalence Class: For a given equivalence relation R in X, equivalence class of a Є X, is denoted by [a], and is
the subset of X containing all elements related to a. i.e. [a] = {b Є X, (a, b) Є R}.
• Function: A function f from a set A to a set B is a specific type of relation for which every element x of set A has
one and only one image y in set B. We write f: A→B, where f(x) = y.
• x is often called the independent variable, and y is called the dependent variable.
• Vertical Line Test: If the graph of the relation is plotted taking all x on x-axis, and all y on y-axis, and a vertical
line which cuts the graph in more than 1 point exists, then the relation is not a function.
• Set A is the domain and Set B is called the codomain of function f.
• The range of the function is the set of images.
• A real function has the set of real numbers or one of its subsets both as its domain and as its range.
• Algebra of functions:
For functions f : X → R and g : X → R, we have
o (f + g) (x) = f (x) + g(x), x Є X
o (f – g) (x) = f (x) – g(x), x Є X
o (f . g) (x) = f (x) . g (x), x Є X
o (k. f) (x) = k ( f (x) ), x Є X, where k is a real number.
o (f / g) (x) = f (x) / g (x), x Є X, g(x) 0
Hg
Name: Class 11 Date:
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
• Classification of functions:
o One to One (or Injective): If 1 element of set A maps to 1 element of set B. e.g. f(x) = x + 1
o Many to One: If more than 1 element of set A maps to 1 element of set B. e.g. f(x) = x
2
o Into: If set B consists of at-least 1 element, which doesn’t have any pre-image in set A. e.g. f(x) = x
2
In other words, Range ⊆ Codomain.
o Onto (or Surjective): If all elements of set B, has a pre-image in A. e.g. f(x) = x + 1
In other words, Range = Codomain
o One to One and Onto (or Bijective): A function is both one to one and onto. e.g. f(x) = x + 1
• Standard Functions:
Function
Graph
Function
Graph
Constant Function
y = c
Domain: R; Range: {c}
Exponential Function
y = e
x
Domain: R; Range = R
+
Linear Function
y = ax + b
Domain: R; Range: R
y = x is called an Identity
function
Logarithmic Function
y = ln x
Domain: R
+
; Range: R
Quadratic Function
y = ax
2
+ bx + c
Domain: R; Range: R
+
+ {0} for
y = x
2
If a = 1, b = c = 0, then we get
the simplest quadratic
function y = x
2
Greatest Integer
Function or step
function:
y = [x]
Domain: R; Range: Z
Cubic Function
y = ax
3
+ bx
2
+ cx + d
Domain: R; Range: R
y = x
3
is the simplest cubic
function
Sine function
y = sin x
Domain: R;
Range: [0,1]
Reciprocal Function
y = 1/x
Domain: R – {0};
Range: R
Cosine function
y = cos x
Domain: R;
Range: [0,1]
Modulus or Absolute Value
Function
y = |x|
Domain: R;
Range: R
+
+ {0}
Tan function
y = tan x
Domain: R-{(2n+1)/2};
Range: R
