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

Class 11 Relations and Functions Notes PDF – Download Free Study Material

Use the link below to download the Relations and Functions Class 11 Notes PDF for free. After that, you can view the PDF anytime you desire without internet access. It is very useful for revision and last-minute studies.

Download PDF

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 $R^{-1}$

Functions

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: R → R, 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:

Question 1:

The domain and range of the function f given by $f (x) = 2 - \left |x -5 \right |$ is:

Solution:

Given data: $f(x)=2-|x-5|$
$\& \mathrm{f}(\mathrm{x})$ is defined by $x \in R$
Thus, its domain is $f(x)=R$

$\begin{aligned}
& |x-5| \geq 0 \\
& -|x-5| \leq 0 \\
& 2-|x-5| \leq 2
\end{aligned}$

Thus, $f(x) \leq 2$
Thus, range of $f(x)=(-\infty, 2]$

Hence, the answer is $(-\infty, 2]$.

Question 2:
Let n(A) = m, and n(B) = n. Then the total number of non-empty relations that can be defined from A to B is:

Solution:

Given data: n(A) = m & n(B) = n

Thus, n(A × B) = n(A).n(B) = mn

Thus, $2^{mn-1}$ is the total no. of relations,

Hence, the answer is $2^{mn-1}$.

Question 3:
If $f(x)=a x+b$, where $a$ and b are integers, $\mathrm{f}(-1)=-5$ and $\mathrm{f}(3)=-3$, then a and b are equal to

Solution:

Given data: $f(x)=a x+b$
Now, $f(-1)=a(-1)+b$
i.e., $-5=-a+b$

Thus, $a-b=5$
Now, $f(3)=3 a+b$
i.e., $3=3 a+b$

Thus, $3 a+b=3$
From (i) & (ii), we get,

$a=2 $ and $ b=-3$

NCERT Class 11 Maths Notes – Chapter-Wise Links

For students' preparation, Careers360 has gathered all Class 11 Maths NCERT notes here for quick and convenient access.

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 NCERT syllabus before making a study routine. The following links will help them check the syllabus. Also, here is access to more reference books.

• NCERT Books Class 11 Maths
• NCERT Syllabus Class 11 Maths
• NCERT Books Class 11
• NCERT Syllabus Class 11