Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, opp.Janakpuri. Ph 9818084221 1
POINTS TO
REMEMBER IN CLASS
XII MATHEMATICS
By
Balraj Khurana
INDEX
1. Relations and functions - Pg 2
2. Inverse trigonometric functions - Pg 5
3. Calculus identities - Pg 6
4. Continuity - Pg 7
5. Differentiation - Pg 8
6. Application of derivative - Pg 9
7. Indefinite integral - Pg 11
8. Definite integral - Pg 14
9. Matrices - Pg 16
10. Determinants - Pg 19
11. Solution of system of linear equations - Pg 21
Brilliant Academy, 303, 3rd Flr, Aggarwal Tower, sec5 Market, Dwarka, Ph 9015048802
, Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, opp.Janakpuri. Ph 9818084221 2
RELATIONS AND FUNCTIONS
I. RELATION
i. Let A and B be two sets. A relation between A and B is a collection of ordered pairs (a, b) such that a
A and b B
ii. If 𝑅: 𝐴 → 𝐵 is a relation from A to B, then 𝑅 ⊆ 𝐴 × 𝐵
iii. If n(A) = m, n(B) = n ,then total number of relations from A to B is 2mn.
iv. Domain of R = {𝑎: (𝑎, 𝑏) ∈ 𝑅}
v. Range of R = {𝑏: (𝑎, 𝑏) ∈ 𝑅}
vi. Co-domain of R = 𝐵
II. Equivalence Relation
Let S be a set and R a relation between S and itself. We call R an equivalence relation on S if R has the
following three properties:
Reflexivity: Every element of S is related to itself ⟹ (𝑎, 𝑎) ∈ 𝑅 ∀ 𝑎 ∈ 𝑆.
Symmetry: If a is related to b then b is related to a . (𝑎, 𝑏) ∈ 𝑅 ⟹ (𝑏, 𝑎) ∈ 𝑅 ∀ 𝑎, 𝑏 ∈ 𝑆.
Transitivity: If a is related to b and b is related to c, then a is related to c. (𝑎, 𝑏) ∈ 𝑅 , (𝑏, 𝑐) ∈ 𝑅 ⟹
(𝑎, 𝑐) ∈ 𝑅 ∀ 𝑎, 𝑏, 𝑐 ∈ 𝑆.
Antisymmetric - A relation is antisymmetric if a R b and b R a⟹ a=b for all values a and b.
III. FUNCTIONS :
Definition - Any relation on A x B in which
i. No two second elements have a common first element and
ii. Every first element has a corresponding second element is called a function. It is also called mapping.
A function is said to map an element x in its domain to an element y in its range. 𝑓: 𝐴 →
𝐵 𝑜𝑟 𝑓: 𝑥 → 𝑓(𝑥) 𝑡ℎ𝑒𝑛 𝑓(𝑥) = 𝑦 where y is a function of x.
DOMAIN - The set of all the first elements of the ordered pairs of a function is called the domain
RANGE - The set of all the second elements of the ordered pairs of a function is called the range
CODOMAIN - If (a, b) is an ordered pair of the function 𝑓: 𝐴 → 𝐵 then the set B is called the Co-Domain. The
range is a subset of the co-domain.
IV. Some important facts about a function from A to B:
Brilliant Academy, 303, 3rd Flr, Aggarwal Tower, sec5 Market, Dwarka, Ph 9015048802
, Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, opp.Janakpuri. Ph 9818084221 3
Every element in A is in the domain of the function; that is, every element of A is mapped to some
element in the range. (If some element in S has no mapping (arrow), then the relation is not a function!)
No element in the domain maps to more than one element in the range.
The mapping is not necessarily onto; some elements of T may not be in the range.
The mapping is not necessarily one-one; some elements of T may have more than one element of S
mapped to them.
S and T need not be disjoint.
V. Types of functions
Injections A function f from A to B is called one to one (or one- one) if whenever 𝒇(𝒙𝟏 ) = 𝒇(𝒙𝟐 ) ⟹
𝒙𝟏 = 𝒙𝟐 . 𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 ℎ𝑒𝑟𝑒 𝑛(𝐴) ≤ 𝑛(𝐵).
Surjections. A function f from A to B is called onto if for all b in B there is an a in A such that f(a) = b.
⟹ ∀𝑦 ∈ 𝐵, ∃𝑥 ∈ 𝐴 ∶ 𝑓(𝑥) = 𝑦. 𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 ℎ𝑒𝑟𝑒 𝑛(𝐴) ≥ 𝑛(𝐵). Range = Co-domain.
Bijections are functions that are injective and surjective i.e. a function f from A to B is called a bijection
if it is one to one and onto.𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 ℎ𝑒𝑟𝑒 𝑛(𝐴) = 𝑛(𝐵)
VI. Some special functions with their domain, range and nature
1. Polynomial function p(x) = a0 + a1x+a2x2+…+anxn ; domain = R; range = R ; continuous
2. Constant Function f(x) = k domain = r ; range = {k} ; continuous
3. Identity function I(x) = x ; domain = R; range = R ; continuous
4. Exponential function f(x) = ex or ax domain = R; domain = (0, ∝) ; continuous
5. Logarithmic function f(x) = logx or In x domain = (0, ∝) : range = R ; continuous
6. Square root function f(x) = √𝑥 ; domain = (0, ∝) ; range = (0, ∝) ; continuous.
7. Sine function - sin: R→ [−1,1]; 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
8. Cosine function - cos: R→ [−1,1]; 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
(2𝑛+1)𝜋
9. Tangent function - tan: R− {𝑥: 𝑥 = } → 𝑅; continuous in its domain
2
(2𝑛+1)𝜋
10. Secant function - sec: R− {𝑥: 𝑥 = } → 𝑅 − (−1,1); continuous in its domain
2
11. Cosecant function - cosec: R−{𝑥: 𝑥 = 𝑛𝜋, 𝑛 ∈ 𝑍 } → 𝑅 − (−1,1); continuous in its domain
12. Cotangent function - cot: R−{𝑥: 𝑥 = 𝑛𝜋, 𝑛 ∈ 𝑍} → 𝑅; continuous in its domain
13. 𝐹𝑙𝑜𝑜𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 x = Greatest integer that is less than or equal to x. domain= R, range = Z;
discontinuous.
Brilliant Academy, 303, 3rd Flr, Aggarwal Tower, sec5 Market, Dwarka, Ph 9015048802
, Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, opp.Janakpuri. Ph 9818084221 4
14. Ceiling function x = Least integer that is greater than or equal to x.domain= R; range = Z;
discontinuous
1
15. Reciprocal function f(x) = 𝑥 ; domain = R - {o};range = R - {o} continuous in R+ and R-
𝑥 , 𝑖𝑓 𝑥 ≥ 0
16. Modulus function f(x) = |𝑥| = { ; Domain = R; Range = R + ; continuous.
−𝑥, 𝑖𝑓 𝑥 < 0
|𝑥| 1, 𝑥 >0
, ∀𝑥 ≠ 0
17. Signum function f(x) = { 𝑥 = { 0 , 𝑥 = 0 ; domain = R ;range = {-1 , 0 ,1}; discontinuous.
0 , 𝑥=0 −1, 𝑥 < 0
VII. COMPOSITION OF FUNCTIONS - function composition is the application of one function to the
results of another. For instance, the functions f: X → Y and g: Y → Z can be composed by computing
the output of g when it has an input of f(x) instead of x. A function g ∘ f: X → Z defined by (g ∘ f)(x) =
g(f(x)) for all x in X.
The composition of functions is always associative. That is, if f, g, and h are three functions with suitably
chosen domains and codomains, then f ∘ (g ∘ h) = (f ∘ g) ∘ h,
The functions g and f are said to commute with each other if g ∘ f = f ∘ g.
VIII. INVERSE OF A FUNCTION - Let ƒ be a bijective function whose domain is the set X, and whose
range is the set Y. Then, if it exists, the inverse of ƒ is the function ƒ–1 with domain Y and range X,
defined by the following rule:
A function with a codomain is invertible if and only if it is both one-to-one and onto or a bijection and
has the property that every element y ∈ Y corresponds to exactly one element x ∈ X.
Domain (f) = range(f-1) and range (f) = domain (f-1)
Inverses and composition - If ƒ is an invertible function with domain X and range Y, then
There is a symmetry between a function and its inverse. Specifically, if the inverse of ƒ is ƒ–1, then the
inverse of ƒ–1 is the original function ƒ. i.e. If 𝑓 −1 ∘ 𝑓(𝑥) = 𝐼𝑋 then 𝑓 ∘ 𝑓 −1 (𝑦) = 𝐼𝑌
Only one-to-one functions have a unique inverse.
If the function is not one-to-one, the domain of the function must be restricted so that a portion of the
graph is one-to-one. You can find a unique inverse over that portion of the restricted domain.
The domain of the function is equal to the range of the inverse. The range of the function is equal to the
domain of the inverse.
Brilliant Academy, 303, 3rd Flr, Aggarwal Tower, sec5 Market, Dwarka, Ph 9015048802
POINTS TO
REMEMBER IN CLASS
XII MATHEMATICS
By
Balraj Khurana
INDEX
1. Relations and functions - Pg 2
2. Inverse trigonometric functions - Pg 5
3. Calculus identities - Pg 6
4. Continuity - Pg 7
5. Differentiation - Pg 8
6. Application of derivative - Pg 9
7. Indefinite integral - Pg 11
8. Definite integral - Pg 14
9. Matrices - Pg 16
10. Determinants - Pg 19
11. Solution of system of linear equations - Pg 21
Brilliant Academy, 303, 3rd Flr, Aggarwal Tower, sec5 Market, Dwarka, Ph 9015048802
, Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, opp.Janakpuri. Ph 9818084221 2
RELATIONS AND FUNCTIONS
I. RELATION
i. Let A and B be two sets. A relation between A and B is a collection of ordered pairs (a, b) such that a
A and b B
ii. If 𝑅: 𝐴 → 𝐵 is a relation from A to B, then 𝑅 ⊆ 𝐴 × 𝐵
iii. If n(A) = m, n(B) = n ,then total number of relations from A to B is 2mn.
iv. Domain of R = {𝑎: (𝑎, 𝑏) ∈ 𝑅}
v. Range of R = {𝑏: (𝑎, 𝑏) ∈ 𝑅}
vi. Co-domain of R = 𝐵
II. Equivalence Relation
Let S be a set and R a relation between S and itself. We call R an equivalence relation on S if R has the
following three properties:
Reflexivity: Every element of S is related to itself ⟹ (𝑎, 𝑎) ∈ 𝑅 ∀ 𝑎 ∈ 𝑆.
Symmetry: If a is related to b then b is related to a . (𝑎, 𝑏) ∈ 𝑅 ⟹ (𝑏, 𝑎) ∈ 𝑅 ∀ 𝑎, 𝑏 ∈ 𝑆.
Transitivity: If a is related to b and b is related to c, then a is related to c. (𝑎, 𝑏) ∈ 𝑅 , (𝑏, 𝑐) ∈ 𝑅 ⟹
(𝑎, 𝑐) ∈ 𝑅 ∀ 𝑎, 𝑏, 𝑐 ∈ 𝑆.
Antisymmetric - A relation is antisymmetric if a R b and b R a⟹ a=b for all values a and b.
III. FUNCTIONS :
Definition - Any relation on A x B in which
i. No two second elements have a common first element and
ii. Every first element has a corresponding second element is called a function. It is also called mapping.
A function is said to map an element x in its domain to an element y in its range. 𝑓: 𝐴 →
𝐵 𝑜𝑟 𝑓: 𝑥 → 𝑓(𝑥) 𝑡ℎ𝑒𝑛 𝑓(𝑥) = 𝑦 where y is a function of x.
DOMAIN - The set of all the first elements of the ordered pairs of a function is called the domain
RANGE - The set of all the second elements of the ordered pairs of a function is called the range
CODOMAIN - If (a, b) is an ordered pair of the function 𝑓: 𝐴 → 𝐵 then the set B is called the Co-Domain. The
range is a subset of the co-domain.
IV. Some important facts about a function from A to B:
Brilliant Academy, 303, 3rd Flr, Aggarwal Tower, sec5 Market, Dwarka, Ph 9015048802
, Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, opp.Janakpuri. Ph 9818084221 3
Every element in A is in the domain of the function; that is, every element of A is mapped to some
element in the range. (If some element in S has no mapping (arrow), then the relation is not a function!)
No element in the domain maps to more than one element in the range.
The mapping is not necessarily onto; some elements of T may not be in the range.
The mapping is not necessarily one-one; some elements of T may have more than one element of S
mapped to them.
S and T need not be disjoint.
V. Types of functions
Injections A function f from A to B is called one to one (or one- one) if whenever 𝒇(𝒙𝟏 ) = 𝒇(𝒙𝟐 ) ⟹
𝒙𝟏 = 𝒙𝟐 . 𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 ℎ𝑒𝑟𝑒 𝑛(𝐴) ≤ 𝑛(𝐵).
Surjections. A function f from A to B is called onto if for all b in B there is an a in A such that f(a) = b.
⟹ ∀𝑦 ∈ 𝐵, ∃𝑥 ∈ 𝐴 ∶ 𝑓(𝑥) = 𝑦. 𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 ℎ𝑒𝑟𝑒 𝑛(𝐴) ≥ 𝑛(𝐵). Range = Co-domain.
Bijections are functions that are injective and surjective i.e. a function f from A to B is called a bijection
if it is one to one and onto.𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 ℎ𝑒𝑟𝑒 𝑛(𝐴) = 𝑛(𝐵)
VI. Some special functions with their domain, range and nature
1. Polynomial function p(x) = a0 + a1x+a2x2+…+anxn ; domain = R; range = R ; continuous
2. Constant Function f(x) = k domain = r ; range = {k} ; continuous
3. Identity function I(x) = x ; domain = R; range = R ; continuous
4. Exponential function f(x) = ex or ax domain = R; domain = (0, ∝) ; continuous
5. Logarithmic function f(x) = logx or In x domain = (0, ∝) : range = R ; continuous
6. Square root function f(x) = √𝑥 ; domain = (0, ∝) ; range = (0, ∝) ; continuous.
7. Sine function - sin: R→ [−1,1]; 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
8. Cosine function - cos: R→ [−1,1]; 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
(2𝑛+1)𝜋
9. Tangent function - tan: R− {𝑥: 𝑥 = } → 𝑅; continuous in its domain
2
(2𝑛+1)𝜋
10. Secant function - sec: R− {𝑥: 𝑥 = } → 𝑅 − (−1,1); continuous in its domain
2
11. Cosecant function - cosec: R−{𝑥: 𝑥 = 𝑛𝜋, 𝑛 ∈ 𝑍 } → 𝑅 − (−1,1); continuous in its domain
12. Cotangent function - cot: R−{𝑥: 𝑥 = 𝑛𝜋, 𝑛 ∈ 𝑍} → 𝑅; continuous in its domain
13. 𝐹𝑙𝑜𝑜𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 x = Greatest integer that is less than or equal to x. domain= R, range = Z;
discontinuous.
Brilliant Academy, 303, 3rd Flr, Aggarwal Tower, sec5 Market, Dwarka, Ph 9015048802
, Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, opp.Janakpuri. Ph 9818084221 4
14. Ceiling function x = Least integer that is greater than or equal to x.domain= R; range = Z;
discontinuous
1
15. Reciprocal function f(x) = 𝑥 ; domain = R - {o};range = R - {o} continuous in R+ and R-
𝑥 , 𝑖𝑓 𝑥 ≥ 0
16. Modulus function f(x) = |𝑥| = { ; Domain = R; Range = R + ; continuous.
−𝑥, 𝑖𝑓 𝑥 < 0
|𝑥| 1, 𝑥 >0
, ∀𝑥 ≠ 0
17. Signum function f(x) = { 𝑥 = { 0 , 𝑥 = 0 ; domain = R ;range = {-1 , 0 ,1}; discontinuous.
0 , 𝑥=0 −1, 𝑥 < 0
VII. COMPOSITION OF FUNCTIONS - function composition is the application of one function to the
results of another. For instance, the functions f: X → Y and g: Y → Z can be composed by computing
the output of g when it has an input of f(x) instead of x. A function g ∘ f: X → Z defined by (g ∘ f)(x) =
g(f(x)) for all x in X.
The composition of functions is always associative. That is, if f, g, and h are three functions with suitably
chosen domains and codomains, then f ∘ (g ∘ h) = (f ∘ g) ∘ h,
The functions g and f are said to commute with each other if g ∘ f = f ∘ g.
VIII. INVERSE OF A FUNCTION - Let ƒ be a bijective function whose domain is the set X, and whose
range is the set Y. Then, if it exists, the inverse of ƒ is the function ƒ–1 with domain Y and range X,
defined by the following rule:
A function with a codomain is invertible if and only if it is both one-to-one and onto or a bijection and
has the property that every element y ∈ Y corresponds to exactly one element x ∈ X.
Domain (f) = range(f-1) and range (f) = domain (f-1)
Inverses and composition - If ƒ is an invertible function with domain X and range Y, then
There is a symmetry between a function and its inverse. Specifically, if the inverse of ƒ is ƒ–1, then the
inverse of ƒ–1 is the original function ƒ. i.e. If 𝑓 −1 ∘ 𝑓(𝑥) = 𝐼𝑋 then 𝑓 ∘ 𝑓 −1 (𝑦) = 𝐼𝑌
Only one-to-one functions have a unique inverse.
If the function is not one-to-one, the domain of the function must be restricted so that a portion of the
graph is one-to-one. You can find a unique inverse over that portion of the restricted domain.
The domain of the function is equal to the range of the inverse. The range of the function is equal to the
domain of the inverse.
Brilliant Academy, 303, 3rd Flr, Aggarwal Tower, sec5 Market, Dwarka, Ph 9015048802
