# CHSE Odisha Class 11 Math Notes Chapter 3 Relations And Function

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 3 Relations And Function will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 3 Relations And Function

Order Pairs
An ordered pair consists of a pair of objects, or elements or numbers or functions in order.
We denote order pairs as (a, b)

• An order pair is not a set of two objects.
• (a, b) = (c, d) ⇒ a = c and b = d
• (a, b) × (b, a)

Cartesian Product Of Sets:
If A and B are non-empty sets, then their Cartesian product, denoted by A × B and defined by A × B = {(a, b): a ∈ A, b ∈ B} = Set of all ordered pairs (a,b) where a ∈ A and b ∈ B
Note:
1. For finite sets A and B |A × B| = |A| . |B|
2. A × B = Φ ⇔ A = Φ or B = Φ
3. A2 = A × A

Properties of Cartesian product:
1. A × B ≠ B × A (Cartesian product is non-commutative)
2. A × (B ∪ C) = (A × B) ∪ (A × C)
3. A × (B ∩ C) = (A × B) ∩ (A × C)
4. A × B = B × A ⇔ A = B
5. A × (B – C) = ( A × B) – (A × C)
6. A ⊂ B ⇒ A × A ⊂ (A × B) ∩ (B × A)
7. A ⊂ B ⇒ A × C ⊂ B × C
8. A ⊂ Band C ⊂ D ⇒ A × C ⊂ B × D
9. (A × B) ∩ (C × D) = ( A ∩ C) × (B ∩ D)

Relation
Let A and B be two arbitrary sets. A binary relation from A to B is a subset of A × B.
OR f is a relation from A to B if f ⊆ A × B
Note:

• If a of A is related to b of B by relation ‘f’ then we write (a,b) ∈ f or a f b
• As Φ ⊂ A × B we have Φ is a relation from A to B. This relation is known as a null of void or empty relation.
• As A × B ⊆ A × B, A × B is also a relation from A to B. This relation is known as universal relation.
• If |A| = m and |B| = n then number of relations from A to B is 2mn

Domain, co-domain, and Range of a relation:
Let f is a relation from A to B. Domain of f = Dom (f) or Df
={x ∈ A : (x, y) ∈ f for some y ∈ B) Co-domain of f = B
Range of ‘f’ = Rng (f) or Rf = {y ∈ B : (x, y) ∈ f for some x ∈ A}

Types Of Relation:
(a) One-many relation: A relation f from A to B is one many if (a, b) and (a, b’) ∈ f ⇒ b ≠ b’
(b) Many-one relations: A relation f from A to B is many-one if (a, b) and (a’, b) ∈ f ⇒ a ≠ a’
(c) One-one relation: A relation f from A to B is one-one if (a, b), (a, b’) ∈ f ⇒ b = b’ and (a, b), (a’, b) ∈ f ⇒ a = a’

Inverse of a relation: Let f is a relation from A to B. The inverse of f is denoted by f-1 is a relation from B to A defined as f-1 = {(b, a): (a, b) ∈ f}

Function:
A relation ‘f’ from X to Y is called a function if:
(a) Df = Dom (f) = X and
(b) (x, y) and (x, z) ∈ f ⇒ y = z or A relation from A to B is a function
if ⇒ Domain of f = X i.e All elements of X is engaged in the relation and
⇒ f is not one many.

Note:
(1) If a relation f from X to Y becomes a function then we write f: X → Y.
(2) If f is a function from A to B i.e. f: X → Y and (x, y) ∈ f then we write y = f(x)
(3) Mapping, map, transformation, transform, operator, and correspondence are different synonym terms of function.
(4) If f: X → Y is defined as y = f(x), then

• y is called the value of the function at x or the image of x under f or the dependent variable.
• x is called the independent variable or pre-image of y under f.

Domain, Co-domain or Range of a function:
Let f: X → Y defined as y = f(x)
(a) Domain of ‘f’ = Dom f = Df = {x ∈ X: y = f(x)}
(b) Range of f = Rng f = Rf = f(A) = {f(x) ∈ Y: x ∈ A } Clearly f(A) ⊆ y
(c) If |A| = m, |B| = n then number of functions from A to B = nm

Real valued function :
A function f: A → B is a real-valued function if B ⊆ R.
→ f is a real function if A ⊆ R and B ⊆ R

Techniques to find Domain and Range of a Real function:
(a) Techniques to find Domain: Let the function is defined as y = f(x).
Step -1: Check the values of x for which f(x) is well defined.
Step -2: The set of all values obtained from step -1 is the domain of ‘f.

(b) Techniques to find range: Let the function is y = f(x)

• Method-1 (By inspection):
→ Step -1: Get values of y for all values x ∈ Dom f.
→ Step -2: Set of all these values of y = Rng f
• Method-2:
→ Step -1: Write x in terms of y
→ Step -2: Get values of y for which x is well defined in Dom f.
→ Step -3: Rng (f) = The set of all y obtained from step 2.

Some Real Functions:
(a) Constant function: A function f: A → R defined as f(x) = k, for some k ∈ R is called a constant function.

(b) Identity function: Let A ⊆ R. The function f: A → A defined as f(x) = x, x ∈ A is called the identity function on A. We denote it by IA

(c) Polynomial function: A function f: A → R defined by f(x) = f(x) = a0 + a1x + a2x2 + anxn where a0, a1, a2, ….., an are real numbers and an ≠ 0 is called a polynomial function (polynomial) of degree n.

(d) Rational function: A function of form f(x) = $$\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}$$ where P(x) and Q(x) are polynomial functions of x is known as a rational function.

(e) Absolute value function OR modulus function: The function f: R → R defined as  f(x) = |x| = $$\begin{cases}x, & x \geq 0 \\ -x, & x<0\end{cases}$$ is called as the modulus function.
→ Rng f = [ 0, ∞] = R+U {0}

Properties Of Modulus Function:
1. For any real number x, we have $$\sqrt{x^2}=|x|$$
2. If a and b are positive real numbers then

• x2 ≤ a2 ⇔ |x| ≤ a
• x2 ≥ a2 ⇔ |x| ≥ a
• a2 ≤ x2 ≤ b2 ⇔ a ≤ |x| ≤ b ⇔ x ∈ [-b, – a] ∪ [a, b]

(f) Signum function: The function f: R → R defined as f(x) = $$\begin{cases}\frac{x}{|x|}, & x \neq 0 \\ 0, & x=0\end{cases}$$ is called signum function.
→ We denote a signum function as f(x) = sgn(x)
→ Range of a signum function = {-1, 0, 1}

(g) Greatest integer function: The function f: R → R defined by f(x) = [x] is called the greatest integer function. [x] = The greatest among all integers ≤ x. OR [x] = n for n ≤ x < n + 1

Properties of the greatest integer function :
Let n is an integer and x is a real number between n and n + 1
(i) [-n]= -[n]
(ii) [x + k] = [x] + k (for an integer ‘k’)
(iii) [-x] = -[x] – 1
(iv) [x] + [-x] = $$\begin{cases}-1, & \mathrm{x} \notin \mathrm{Z} \\ 0, & \mathrm{x} \in \mathrm{Z}\end{cases}$$
(v) [x] – [-x] = $$\begin{cases}2[\mathrm{x}]+1, & \mathrm{x} \notin \mathrm{Z} \\ 2[\mathrm{x}], & \mathrm{x} \in \mathrm{Z}\end{cases}$$
(vi) [x] ≥ k ⇒ x ≥ k for k ∈ Z
(vii) [x] ≤ k ⇒ x < k +1 for k ∈ Z
(viii) [x] > k ⇒ x > k + 1 for k ∈ Z
(ix) [x] < k ⇒ x < k for k ∈ Z

(h) Exponential Function: A function f: R → R defined as f(x) = ax where a > 0 and a ≠ 1 is called the exponential function.

Properties Of Exponential Function:
1. ax+y =  ax . ay
2. (ax)y = axy
3. ax = 1 if x = 0
4. If a > 1, ax > ay ⇒ x > y
5. If a < 1, ax > ay ⇒ x < y

Logarithmic Function:
Let a ≠ 1 is a positive real number. The function f: (0, ∞) → R defined by f(x) = logax is called the logarithmic function, where y = loga ⇔ ay = x
→ Domain of a logarithmic function = (0, ∞) and Range = R

Properties of logarithmic function:
1. loga (xy) = logax + logay
2. loga (x/y) = logax – logay
3. logaa = 1
4. loga(x)y = y logax
5. loga x = 0 ⇔ x = 1
6. logax = $$\frac{1}{\log _a{ }_a}$$ , x ≠ 1
7. logab = $$\frac{\log _a b}{\log _c a}$$
8. $$\log _{a^n}\left(x^m\right)$$ = $$\frac{m}{n}$$ loga|x|

Different Categories of function:
(a) Algebraic Function: A function that can be generated by a variable by a finite number of algebraic operations such as addition, subtraction, multiplication, division, square root, etc. is called an algebraic function.

(b) Transcendental function: A non-algebraic function is a transcendental function.
⇒ Trigonometric, trigonometric, Exponential, and logarithmic functions are transcendental functions.

Even And Odd Functions:
A function ‘f’ is an even function  if f(-x) = x and is an odd function
if f(-x) = x and is an odd function: if f(-x) = -f(x)
Note:
1. If ‘f’ is any function f(x) + f(-x) is always an even function and f(x) – f(-x) is an odd function.
2. Every function f(x) can be expressed as the sum of an even and an odd function as f(x) = g(x) + h(x), where
g(x) = $$\frac{f(x)+f(-x)}{2}$$
h(x) = $$\frac{f(x)-f(-x)}{2}$$

Periodic Function:
A function is called a periodic function with period k if f(x + k) = f(x) for some constant k ≠ 0. The least positive value of k for which f(x + k) = f(x) holds is called the fundamental period of f.

Properties of periodic function :
(1) If k is the period of f then any non-zero integral multiples of k is also a period of f.
(2) If k is the period of f(x) then f(ax + b) is also periodic with period $$\frac{k}{a}$$
(3) If f1(x) + f2(x) and f3(x) are periodic functions with periods k1, k2, k3, respectively then the function a1f1(x) + a2f2(x) + a3f3(x) is also a periodic function with period, LCM (k1, k2, k3)

Algebra Of Real functions:
(a) Equality of two functions: Two functions f and g are equal iff ‘
(i) Dom f = Dom g
(ii) Co-Dom f = Co-Dom g
(iii) f(x) = g(x) for all x belonging to their common domain.

(b) Addition of two functions: Let f: D1 → R and g: D2 → R be two real functions.
The sum function f + g is defined by f + g: D1 ∩ D2 → R and (f + g)(x) = f(x) + g(x) ∀ x ∈ D1 n D2

(c) Subtraction of two functions: Let f: D1 → R and g: D2 → R. The difference function (f – g) is f – g: D1 ∩ D2) → R defined by (f – g) (x) = f(x) – g(x) ∀ x ∈ D1 ∩ D2

(d) Scalar multiplication: Let f: D → R and c is any scalar. The scalar multiple of f by the scalar c is cf: D → R defined as (cf)(x) = c. f(x) ∀ x ∈ D1.

(e) Multiplication of two functions: Let f: D1 → R and g: D2 → R are two real functions. The product function (fg) is (fg): D1 ∩ D2 → R defined as (fg)(x) = f(x)g(x) ∀ ∈ D1 ∩ D2

(f) The quotient of two functions: Let f: D1 → R and g: D2 → R are two real functions. the quotient function ($$\frac{f}{g}$$) i,e,. $$\frac{f}{g}$$: D1 ∩ D2 → R, defined by ($$\frac{f}{g}$$)(x) = $$\frac{f(x)}{g(x)}$$, ∀ x ∈ D1 ∩ D2