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CBSE NCERT NOTES CLASS 11 MATHS CHAPTER 2

Relations And Functions

Chapter Notes

Ordered Pair

Equality of Ordered Pairs

Cartesian Product

- Properties of Cartesian Product

Relation

Domain

Codomain

Range

Inverse Relation

Function

Equal Functions

Laws of logarithms

Logarithm as inverse function of exponential function

Natural logarithm (ln)

Inverse logarithm (antilog) calculation

Logarithm rules

Algebra of Real Functions

Addition of two real functions

Subtraction Real Functions

Multiplication by a scalar

Multiplication of two real functions

Quotient of two real functions

CBSE NCERT NOTES CLASS 11 MATHS CHAPTER 2

Relations And Functions

Chapter Notes

Ordered Pair

An ordered pair consists of two objects or elements in a given fixed order, e.g., (a,b).

Equality of Ordered Pairs

Two ordered pairs (a1, b1) and (a2, b2) are equal iff a1 = a2 and b1 = b2.

Cartesian Product

Given two non-empty sets P and Q, the cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e.,

P × Q = {(p,q) : p ∈ P, q ∈ Q }

If either P or Q is the null set, then P × Q will also be empty set, i.e.,

P × Q = φ

If there are three sets P, Q, R and p ∈ P, q ∈ Q and r ∈ R, then (p, q, r) is called an ordered triplet. Then,

P × Q × R = {(p, q, r) : p ∈ P, q ∈ Q, r ∈ R }

Properties of Cartesian Product

For three sets A, B and C

• n (A × B)= n(A) n(B)

• A × B = Φ, if either A or B is Φ.

• A × (B ∪ C)= (A × B) ∪ (A × C)

• A × (B ∩ C) = (A × B) ∩ (A × C)

• A × (B - C)= (A × B) -(A × C)

• (A × B) ∩ (C × D)= (A ∩ C) × (B ∩ D)

• If A ⊆ B and C ⊆ D, then (A × C) ⊂ (B × D)

• If A ⊆ B, then A × A ⊆ (A × B) ∩ (B × A)

• A × B = B × A ⇔ A = B

• If either A or B is an infinite set, then A × B is an infinite set.

• A × (B’ ∪ C’)’ = (A × B) ∩ (A × C)

• A × (B’ ∩ C’)’ = (A × B) ∪ (A × C)

• If A and B be any two non-empty sets having n elements in common, then A × B and B × A have n2 elements in common.

• If A ≠ B, then A × B ≠ B × A

• If A = B, then A × B= B × A

• If A ⊆ B, then A × C ⊆ B × C

Relation

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

If R ⊆ A × B and (a, b) ∈ R, then we say that a is related to b by the relation R, written as aRb.

The first element is called pre-image and the second element is called the image of the first element.

Domain

The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R (it is not always A)

Codomain and Range

The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R. Note that range ⊆ codomain.

Thus, domain of R = {a : (a , b) ∈ R} and range of R = {b : (a, b) ∈ R}

• A relation may be represented algebraically either by the Roster method or by the Set-builder method.

• An arrow diagram is a visual representation of a relation.

Inverse Relation

If A and B are two non-empty sets and R be a relation from A to B, such that R = {(a, b) : a ∈ A, b ∈ B}, then the inverse of R, denoted by R-1 , i a relation from B to A and is defined by R-1 = {(b, a) : (a, b) ∈ R}

Function

A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.

In other words, a function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element.

Equal Functions

Two functions f and g are said to be equal iff,

1. Domain of f = domain of g,

2. Co-domain of f = co-domain of g and

3. f(x) = g(x) for all x belonging to co-domain

Real Valued Function and Real Function

A function which has either R or one of its subsets as its range is called a real valued function.

A function which has either R or one of its subsets as its range and domain, it is called a real function.

To find domain of a function

Case 1: If the rational function is of the form $\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)}$. The denominator g(x) should not be zero.

Case 2: If the function involves square roots, then the expression within the square root should not be negative.

Solve the equations and write the answer in interval notation.

To find range of a function

1. Write y = f (x)

2. Solve x in terms of y, let x = g(y)

3. Find the values of y for which the values of x are real and in the domain of f.

4. This set is the range of f (x).

5. In case of complicated functions, it is better to draw graph of the function.

Some Real Valued Functions and Their Graphs

Rational functions are functions of the type

$\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)}$,

where f(x) and g(x) are polynomial functions of x defined in a domain and g(x) ≠ 0.

Congruence Modulo m

Let m be an arbitrary but fixed integer. Two integers a and b are said to be congruence modulo m, if a – b is divisible by m and we write a ≡ b (mod m). i.e., a ≡ b (mod m) ⇔ a – b is divisible by m.

Identity function

A function f : R R defined by y = f(x) = x for each x R, is called the identity function.

Both domain and range are R.

Constant function

The function f : R R defined by y = f (x) = c, x R where c is a constant and each x R. Here domain of f is R and its range is {c}.

Polynomial function

A function f : RR is said to be polynomial function if for each x in R, y = f (x) = a0 + a1x + a2x2 + ...+ an xn, where n is a non-negative integer and a0, a1, a2,...,an ∈R.

Examples, f(x) = x3 - x2 + 2, and g(x) = x4 + $\sqrt{2}$x.

But x3/2 + 2x is not a polynomial function. (Why?)

For f: R R by y = f(x) = x2, x R.

Domain of f = {x : x R}.

Range of f = {x : x ≥ 0, x R}.

Reciprocal Function

f : R – {0} → R defined by f (x) = $\frac{1}{\mathrm{x}}$

Domain = R – {0}, range = R – {0}

The Modulus function

The function f : RR defined by f(x) = |x| for each x R is called modulus function.

Signum function

The function f : RR defined by

is called the signum function.

It can also be defined as,

The domain of the signum function is R and the range is the set {–1, 0, 1}.

Greatest integer function or floor function

The function f: R R defined by f(x) = ⌊x⌋, x R, assumes the value of the greatest integer, less than or equal to x is called the greatest integer function.

⌊x⌋= –1 for –1 ≤ x < 0

⌊x⌋= 0 for 0 ≤ x < 1

⌊x⌋= 1 for 1 ≤ x < 2

⌊x⌋ = 2 for 2 ≤ x < 3 and so on.

Smallest integer function or ceiling function

The function f: R R defined by f(x) = ⌈x⌉, x R, assumes the value of the smallest integer, greater than or equal to x is called the smallest integer function.

x⌉ = 0 for –1< x ≤ 0

x⌉ = 1 for 0 < x ≤ 1

x⌉ = 2 for 1 < x ≤2

x⌉ = 2 for 2 < x ≤ 3 and so on.

Fractional part function

The function f: R R defined by f(x) = {x}, x R, defined as,

{x} = x - ⌊x⌋

is called the fractional part function.

{1.2} = 1.2 - 1 = 0.2

{- 1.2} = -1.2 – (-2) = 0.8

 Square Function: The function f : R → R defined by is called square function. Square Root Function: The function, f : R+ → R defined by is called square root function.

Exponential function

If a is a positive real number, other than unity, then a function that maps each x ∈ R to ax is called exponential function.

For 0 < a < 1, the exponential function is a strictly decreasing function

For a > 1, the exponential function is a strictly increasing function

Logarithmic Function

If a is a positive real number, other than unity, then,

is defined as logarithmic function. b is called the base.

For 0 < b < 1, the exponential function is a strictly decreasing function and is negative

For b > 1, the exponential function is a strictly increasing function and is positive.

Laws of logarithms

If

x = b y,

then

logb (x) = y

For example,

100 = 102 ⇒ log10 (100) = 2

Logarithm as inverse function of exponential function

The logarithmic function,

y = logb(x)

is the inverse function of the exponential function,

Also x = logb (bx)

Natural logarithm (ln)

ln (x) = loge (x)

Inverse logarithm (antilog) calculation

The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y

x = log-1(y) = b y

Logarithm rules

${\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\mathrm{x}\right)=\frac{{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{c}}\left(\mathrm{x}\right)}{{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{c}}\left(\mathrm{b}\right)}$

Algebra of real functions

Addition of two real functions

Let f : X → R and g : X → R be any two real functions, where X ⊂ R.

Then (f + g): X → R is defined by

(f + g) (x) = f (x) + g (x), for all x ∈ X.

Subtraction Real Functions

Let f : X → R and g : X → R be any two real functions, where X ⊂ R.

Then (f + g): X → R is defined by

(f - g) (x) = f (x) - g (x), for all x ∈ X.

Multiplication by a scalar

Let f : X→R be a real valued function and k be a scalar.

Then the product kf is a function from X to R defined by

(kf ) (x) = kf (x), x ∈X.

Multiplication of two real functions

The product (or multiplication) of two real functions f : X→R and g : X→R is a function f g : X→R defined by

(f g) (x) = f (x) g (x), for all x ∈ X.

This is called pointwise multiplication.

Quotient of two real functions

Let f and g be two real functions defined from X→R where X ⊂R. The quotient of f by g denoted by f/g is a function defined by,