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. A^{2} = 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 2
^{mn}

Domain, co-domain, and Range of a relation:

Let f is a relation from A to B. Domain of f = Dom (f) or D_{f}

={x ∈ A : (x, y) ∈ f for some y ∈ B) Co-domain of f = B

Range of ‘f’ = Rng (f) or R_{f }= {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) D_{f} = 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 = D_{f} = {x ∈ X: y = f(x)}

(b) Range of f = Rng f = R_{f} = 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 = n^{m}

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 I_{A}

(c) Polynomial function: A function f: A → R defined by f(x) = f(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{n}x^{n} where a_{0}, a_{1}, a_{2}, ….., a_{n} are real numbers and a_{n} ≠ 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

- x
^{2}≤ a^{2}⇔ |x| ≤ a - x
^{2}≥ a^{2}⇔ |x| ≥ a - a
^{2}≤ x^{2}≤ b^{2}⇔ 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) = a^{x} where a > 0 and a ≠ 1 is called the exponential function.

**Properties Of Exponential Function:**

1. a^{x+y} = a^{x} . a^{y}

2. (a^{x})^{y} = a^{xy}

3. a^{x} = 1 if x = 0

4. If a > 1, a^{x} > a^{y} ⇒ x > y

5. If a < 1, a^{x} > a^{y} ⇒ x < y

**Logarithmic Function:**

Let a ≠ 1 is a positive real number. The function f: (0, ∞) → R defined by f(x) = log_{a}^{x} is called the logarithmic function, where y = log_{a}^{x } ⇔ a^{y} = x

→ Domain of a logarithmic function = (0, ∞) and Range = R

Properties of logarithmic function:

1. log_{a} (xy) = log_{a}x + log_{a}y

2. log_{a} (x/y) = log_{a}x – log_{a}y

3. log_{a}a = 1

4. log_{a}(x)^{y} = y log_{a}x

5. log_{a} x = 0 ⇔ x = 1

6. log_{a}x = \(\frac{1}{\log _a{ }_a}\) , x ≠ 1

7. log_{a}b = \(\frac{\log _a b}{\log _c a}\)

8. \(\log _{a^n}\left(x^m\right)\) = \(\frac{m}{n}\) log_{a}|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 f_{1}(x) + f_{2}(x) and f_{3}(x) are periodic functions with periods k_{1}, k_{2}, k_{3}, respectively then the function a_{1}f_{1}(x) + a_{2}f_{2}(x) + a_{3}f_{3}(x) is also a periodic function with period, LCM (k_{1}, k_{2}, k_{3})

** 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: D_{1} → R and g: D_{2} → R be two real functions.

The sum function f + g is defined by f + g: D_{1} ∩ D_{2} → R and (f + g)(x) = f(x) + g(x) ∀ x ∈ D_{1} n D_{2}

(c) Subtraction of two functions: Let f: D_{1} → R and g: D_{2} → R. The difference function (f – g) is f – g: D_{1} ∩ D_{2}) → R defined by (f – g) (x) = f(x) – g(x) ∀ x ∈ D_{1} ∩ D_{2}

(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 ∈ D_{1}.

(e) Multiplication of two functions: Let f: D_{1} → R and g: D_{2} → R are two real functions. The product function (fg) is (fg): D_{1} ∩ D_{2 }→ R defined as (fg)(x) = f(x)g(x) ∀ ∈ D_{1} ∩ D_{2}

(f) The quotient of two functions: Let f: D_{1} → R and g: D_{2} → R are two real functions. the quotient function (\(\frac{f}{g}\)) i,e,. \(\frac{f}{g}\): D_{1} ∩ D_{2 }→ R, defined by (\(\frac{f}{g}\))(x) = \(\frac{f(x)}{g(x)}\), ∀ x ∈ D_{1} ∩ D_{2}