Relations And Functions Class 11th Notes - Free NCERT Class 11 Maths Chapter 2 notes - Download PDF

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NCERT Class 11 Maths Chapter 2 Notes: Relations and Functions

Understanding how elements connect and interact with each other is at the heart of mathematics, and NCERT Class 11 Maths Chapter 2: Relations and Functions lays the foundation for just that.

Ordered Pair: Two elements form an ordered pair.

Representation: (a,b)

NOTE: 2 ordered pairs (a, b) and (c, d) are said to be equal if (a=c) and (b=d)

Cartesian Product

If sets A and B have an ordered pair of (a,b) where aA and bB are called the Cartesian product.

Denoted by: A x B

Set builder form: A x B ={(a,b) : a ∈A and b∈ B}

NOTE:

1. If A=∅ , B= ∅, then the cartesian product A x B=∅.

2. It doesn’t satisfy the commutative law: (A, B)≠(B, A)

3. If either of the two sets is an infinite set, then the whole product becomes infinite.

A = infinite, B = finite, then A x B and B x A are also infinite.

4. If n(A) = m (m = number of elements in A), n(B) = n, then the number of elements in the Cartesian product is mn.

Eg: n(A) = 3 and n(B) = 2 then n(A x B) = 6.

Relations

A group of ordered pairs containing one element from each set is called a relation between two sets. Suppose A, and B are both non-empty sets, then the relation is a subset of the Cartesian product of (A x B).

A subset is a relation between the first and second elements of ordered pairs in A x B.

The set of first elements in relations is called the domain, and the next element is called the image or range of R.

Set builder form: R={(a,b): (a,b)∈R}

Representation Of Relation

Note:

• The range can be represented in sand et-builder, roaster form, and also using arrow marks, with brackets.
• If n(A) = m (m = number of elthe ements in A), n(B) = n, and cartesian product=mn then a number of relations caound by using $2^{m n}$

Inverse Relation

Sets A and B are both non-empty sets with $R$ being the relation, and the inverse of the relation is called the inverse relation $R^{-1}$ from B to A.

Set builder form : $R^{-1}=\{(b, a):(a, b) \in R\}$

The domain of the relation R will be the range of the inverse relation $R^{-1}$. The range of the relation R will be the domain of the inverse relation $R^{-1}$

Function

A relation F from A to B is called a function if every element in set A of a function has only one image in set B of a 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 such that no two pairs in the relation have the same first element.

The notation $f: \mathrm{X} \rightarrow \mathrm{Y}$ means that $f$ is a function from X to $\mathrm{Y}. \mathrm{X}$ is called the domain of $f$ and Y is called the co-domain of $f$. Given an element $x \in \mathrm{X}$, there is a unique element $y$ in Y that is related to $x$. The unique element $y$ to which $f$ relates $x$ is denoted by $f(x)$ and is called $f$ of $x$, or the value of $\boldsymbol{f}$ at $\boldsymbol{x}$, or the image of $x$ under $f$.

The set of all values of $f(x)$ taken together is called the range of $f$ or image of X under $f$. Symbolically.

range of $f=\{y \in \mathrm{Y} \mid y=f(x)$, for some $x$ in X$\}$

Real-Valued Functions

Function denoted as f: A→B is said to be a real-valued function if B is a subset of R . If A, and B are subsets of R, in such conditions, we can call f a real function.

Type Of Functions:

Identity function:

A function is said to be an identity function if a function f: R→R when f(x)=x for each x belonging to R satisfies.

Consider the graph of an identity function f(x)=x:

Domain= R

Range = C

Constant Function:

A function is said to be a constant function if a function f: R R when f(x)=C for each C belonging to R.

Consider the graph of a constant function:

Domain= R

Range = C

Polynomial Function:

A function is said to be a Polynomial function if a function f: RR when each x belonging to R satisfies.

Rational Functions:

The functions that belong to Real functions, and are represented as f(x)/g(x) where f(x) and g(x) ≠ 0, are polynomial functions that are represented using x, and they belong to R.

Modulus Function:

Real function f: R → R is said to be a modulus function if

$f(x)=|x|$ for all x belongs to R
Domain: $R$
Range : $R^{+} \cup 0$ in the interval $[0, \infty)$

$f(x)=\left\{\begin{array}{cc}-x & \text { if } x<0 \\ x & \text { if } x \geq 0\end{array}\right\}$

Signum Function:

Real function f: R → R is said to be a Signum function if f(x) = lxl / x where x≠0 and when x = 0 we get

$f(x)=\left\{\begin{array}{cc}1, & \text { if } x>0 \\ 0, & \text { if } x=0 \\ -1, & \text { if } x<0\end{array}\right.$

Domain= R

Range={-1,0,1}

Greatest Integer Function:

Real function f: R → R is said to be the greatest integer function if f(x)=[x] where x belongs to R and values of x are the greatest integer or less than or equal to x.

Fractional Function:

Real function f: R → R is said to be a rational function if f(x)={x} where x belongs to R.

f(x) = {x} = x – [x]

Domain: R

Range : [0,1)

Algebra of Real Functions

Addition Of Two Algebraic Functions:

If f: X → R and g: X →R are real functions, then we represent: (f + g): X → R as (f + g) (x) = f (x) + g(x) where x belongs to X.

Subtraction Of Real Functions:

If f: X → R and g: X → R are real functions, then we represent : (f – g): → R as (f – g) (x) = f (x) – g(x) where x belongs to X.

Multiplication Of Scalar:

If f: X → R is a real function, K is any scalar

The product of Kf is defined by : (Kf)(x) = Kf(x)

Multiplication Of Two Real Functions:

If f: → R and g: X→ R are real functions,

The product of the two functions is defined by: (fg) x = f(x). gx)

The Quotient Of Two Real Functions:

If f and g are two real functions, then the

The quotient of f by g is given by fg from XR.

(f/g)(x) = f(x)/g(x) where g(x)≠0

With this topic, we conclude the NCERT Class 11 Maths Chapter 2 Notes.

Relations and Functions: Previous Year Question and Answer

Given below are some previous year question answers of various examinations from the NCERT class 11 chapter 2, Relations and Functions:

NCERT Class 11 Notes Chapter Wise

All the links of chapter-wise notes for NCERT class 11 maths are given below:

Subject-Wise NCERT Exemplar Solutions

After finishing the textbook exercises, students can use the following links to check the NCERT exemplar solutions for a better understanding of the concepts.

• NCERT Exemplar Class 11 Solutions
• NCERT Exemplar Class 11 Maths
• NCERT Exemplar Class 11 Physics
• NCERT Exemplar Class 11 Chemistry
• NCERT Exemplar Class 11x Biology

Subject-Wise NCERT Solutions

Students can also check these well-structured, subject-wise solutions.

• NCERT Solutions for Class 11 Mathematics
• NCERT Solutions for Class 11 Chemistry
• NCERT Solutions for Class 11 Physics
• NCERT Solutions for Class 11 Biology

NCERT Books and Syllabus

Students should always analyse the latest CBSE syllabus before making a study routine. The following links will help them check the syllabus. Also, here is access to more reference books.

• NCERT Book for Class 12
• NCERT Syllabus for Class 12