NCERT Class 12 Maths Chapter 1 Notes, Relations and Functions Class 12 Chapter 1 Notes

Imagine you are hungry and go online to check the prices to order food. You can see there is a particular price to be paid for every type of item. So, here we can see the relationship between food and its prices. Also, we can see the example of a function where one price is associated with one particular item. In Maths chapter 1 class 12, students read Relations and Functions, which is an important concept of Mathematics. Students should already have an idea about the basic concepts of this chapter from class 11. The relations and functions class 12 questions and answers are more advanced and need more practice, as this chapter has good weightage in the CBSE Class 12 Board Exam as well as in the JEE Main Entrance exam.

After going through the textbook exercises and solutions, students need a type of study material from which they can recall concepts in a shorter time. Relations and functions class 12 notes are very handy in this cause. These NCERT notes are prepared by experienced Careers360 experts, comprise a short description of all the topics covered in this chapter. Students can also practice NCERT Exemplar Class 12 Maths Solutions Chapter 1 Relation and Functions to have a firm grasp of this chapter.

Relation:

Two sets, A and B, are related if there is a recognizable connection or link between the elements of sets A and B. Let ‘a’ be an element of set A and ‘b’ be an element of set B. If $(a,b)\in R$, we can say that a is related to b under the relation R. We can write as "aRb" to describe the relation.

Let a relation R be defined as $R:A\to B$.

Example: If A = {1, 2, 3} and B = {a, b, c}, then R = {(1, b), (2, c), (1, a), (3, a)} being a subset of A × B, is a relation from A to B.
Here, (1, b), (2, c), (1, a), and (3, a) belong to R, so we write 1 Rb, 2Rc, 1Ra, and 3Ra.

Total Number of Relations: Let A and B be two non-empty finite sets consisting of m and n elements, respectively. Then A x B consists of mn ordered pairs. So, the total number of subsets of A x B is 2mn.

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.

Domain of $\mathrm{R}\{\mathrm{a}:(\mathrm{a},\mathrm{b})\in \mathrm{R}\}$

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.

Range of $R\{b:(a,b)\in R\}$

Co-domain: The whole set B is called the co-domain of the relation R.

Note- Range subset of co-domain.

Types of Relation:

Empty Relation: No element of $A$ is related to any element of $A$, i.e.,
$\mathrm{R}=\varphi \subset \mathrm{A}\times \mathrm{A}$
Universal relation: Each element of $\mathrm{set}\mathrm{A}$ is related to every element of $\mathrm{set}\mathrm{A}$ i.e.

$\mathrm{R}=\mathrm{A}\times \mathrm{A}$

Note: Both empty relation and universal relation are called trivial relation.

Identity Relation: A relation R on A is called the identity relation if every element of A is related to itself only. Let A be a set. Then the relation R = {(a, a) : a belongs to A} on A is called the identity relation on A.

Reflexive Relation: A relation R on a set A is said to be reflexive if every element of A is related to itself. If $(a,a)\in R$ for every $a\in A$.

Note: Every Identity relation is reflexive, but every reflexive relation is not identity.

Symmetric Relation: if $(a_1,a_2)\in \mathrm{R}\Rightarrow (a_2,a_1)\in \mathrm{R}\text{for all }{a}_1,a_2\in \mathrm{A}$

Transitive Relation: if $(a_1,a_2)\in \mathrm{R}$ and $(a_2,a_3)\in \mathrm{R}\Rightarrow (a_1,a_3)\in \mathrm{R}$ for all $a_1,a_2,a_3\in \mathrm{A}$

Equivalence Relation: If a relation R is reflexive, symmetric, and transitive, then it is said to be an equivalence relation.

Function:

The function is a binary relation between two sets A and B that associates each element of set A to exactly one element of set B. So, If a vertical line cuts a given graph at more than one point then it can not be the graph of a function.

Note - Every function is a relation, but every relation is not necessarily a function.

Let a function 'f' is defined as $f:A\to B$.

Domain: set A is called the domain of the function.

Co-domain: set B is called the co-domain of the function.

Range: The set of 'f' images of all the elements of set A is known as the range of the function 'f'.

Types of Function:-

Injective function: A function $f$ is called injective if the images of distinct elements of $A$ under $f$ are distinct.
If function $f:\mathrm{A}\to \mathrm{B}$ is defined for every $a_1,a_2\in \mathrm{A},f\left(a_1\right)=f\left(a_2\right)$ $\Rightarrow a_1=a_2$. then this is one-one function.

Many-one function: A function f is called many-one if the images of distinct elements of A under f can be the same.

Surjective function: A function $f:X\to Y$ is called Surjective if every element of Y is the image of some element of X.

Composition of Functions and Invertible Function:-

Composite function:-

If function $f:A\to B$ and $g:B\to C$ are two functions then the composite function of two functions 'f' and 'g' is $g\circ f:A\to C$ is defined as $g\circ f(x)=g(f(x))\mathrm{\forall }x\in A$.

Example-

If $f(x)=8x^3$ and $g(x)=x^{\frac{1}{3}}$ than find $\mathrm{f}(\mathrm{g}(\mathrm{x}))$ and $\mathrm{g}(\mathrm{f}(\mathrm{x}))$
Solution:
The solution is as follows

$\begin{array}{rl}& \text{ (ii) }f(x)=8x^3\text{ and }g(x)=x^{\frac{1}{3}}\\ & \text{ g o f }=g(f(x))\\ & =g\left(8x^3\right)\\ & ={\left(8x^3\right)}^{\frac{1}{3}}\\ & =2x\end{array}$

$\begin{array}{rl}& \text{f o g}=f(g(x))\\ & =f\left(x^{\frac{1}{3}}\right)\\ & =8\left({\left(x^{\frac{1}{3}}\right)}^3\right)\\ & =8x\end{array}$

Inverse function:-
If function $f:\mathrm{A}\to \mathrm{B}$ be a one-one and onto function, then there exists a unique function $g:\mathrm{B}\to \mathrm{A}$ such that
$f(x)=y\iff g(y)=x$ $\mathrm{\forall }x\in A\mathrm{\&}y\in B$.
The function g is the inverse of f.

$g={\mathrm{f}}^{-1}:\mathrm{B}\to \mathrm{A}.$

Example:
If the function $f(x)=4x+3$ is defined $f:R\to R,$ then find the inverse of $f$.
Solution:

$\begin{array}{rl}& f(x)=4x+3\\ & y=4x+3,y\in R\\ & \Rightarrow x=f^{-1}(y)=\frac{y-3}{4}\in R\end{array}$

Algebraic Operations on Functions:-

Let two real-valued functions f & g be given with domain set A, B respectively.

(i) (f + g ) ( x) = f(x) + g(x) (domain is $A\cap B$ )

(ii) (f - g ) ( x) = f(x) - g(x) (domain is $A\cap B$ )

(iii) (f . g ) ( x) = f(x) g(x) (domain is $A\cap B$ )

(iv) $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$ (domain is $A\cap B$ - { x | g(x) = 0 } )

Binary Operations:-
Binary operation * on a set A is function $\ast :A\times A\to A$. Binary operation * is denoted $\ast (a,b)$ as $a\ast b$.

Commutative property:- If $a\ast \mathrm{b}=\mathrm{b}\ast \mathrm{a}$ for every $\mathrm{a},\mathrm{b}\in \mathrm{A},\text{than * }$ is commutative on A.

Associative property:- If $(a\ast \mathrm{b})\ast c=a\ast (\mathrm{b}\ast \mathrm{c})$ for every $\mathrm{a},\mathrm{b},c\in \mathrm{A},\text{than }$ * is associative on $A$.

Identity:- If an element e in set A exists such that $a\ast \mathrm{e}=\mathrm{a}=\mathrm{e}\ast \mathrm{a},\mathrm{\forall }\mathrm{a}\in \mathrm{A}$ than ' e ' is an identity element for * on set A.

Note- Zero is the identity for addition operation on R but not on N as 0 doesn't belong to N.

NCERT Class 12 Notes Chapter Wise

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 clarity of the concepts.

• NCERT Exemplar Class 12 Solutions

• NCERT Exemplar Class 12 Maths

• NCERT Exemplar Class 12 Physics

• NCERT Exemplar Class 12 Chemistry

• NCERT Exemplar Class 12 Biology

Subject Wise NCERT Solutions

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

• NCERT Solutions for Class 12 Mathematics

• NCERT Solutions for Class 12 Chemistry

• NCERT Solutions for Class 12 Physics

• NCERT Solutions for Class 12 Biology

NCERT Books and Syllabus

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

Relations and Functions Class 12 Notes: Important Points to Remember

Before assessing the notes, students should follow these following points.

• Try to solve all the NCERT exercises on your own. You can take help from these CBSE Class 12 Maths chapter 1 notes, which will help you to understand the concepts easily.

• After solving the NCERT problems, students should solve the CBSE 12 Board Previous Year's Papers to have an idea about the important types of questions.

• Class 12 Maths chapter 1 notes PDFs are always handy for offline quick revision before the exam.