Relations and Functions Class 11 Maths – NCERT Solutions

It’s time to read – “Class 11 Maths Chapter 2 Relations and Functions”. Class 11 Maths Chapter Relation and function are based on chapter 1 Sets of class 11 Maths. I don’t Know anything about sets, You should read the Sets Before Relations and Functions.

Sets Chapter is the basis of relations and also functions of Class 11th. First of all, You know “What do you mean by Sets”. Sets Chapter is 1st Chapter and relations and functions are 2nd Chapter of Class 11th Maths. Now, it’s time to learn about Relation not Sets so if you want to learn more about the sets chapter of class 11th maths.

You can also learn the Class 11th Maths Chapter 1 Sets.

What do you mean by relation? Describe it with an example in brief.

Relation – It represents the relationship between two different sets. It means the cartesian products of sets is represent with relation.

Example of Relation – A = {x : y, x is brother of y}, B = {a : b, a = 2b + 5}, C = (x : y, x is classmate of y}, D = { a:b, a² = 4b² + 6}, E = {a:b, a is sister of b}.

I hope, you understand “what is a relation” with these examples. We learn the method of representing relations as we know the method of representing set is two: roaster method and tabular method. Similarly, the method of representing relation is the same assets and the same as functions.

Now, you will learn types of relations in brief with examples.

In class 11 Maths NCERT Book, Relations and functions are 2nd chapter but in class 12 Maths NCERT Book, Relations and functions are 1st chapter.
You can also learn the NCERT Class 12 Maths Chapter 1 Relation and functions.

1. Introduction

In mathematics:

• Relation: A relation between two sets is a rule that assigns elements of one set to elements of another set.
• Function: A function is a special type of relation in which each element of the first set (domain) is related to exactly one element of the second set (codomain).

Why it matters:
Understanding relations and functions is crucial for graphing, transformations, and advanced mathematics in Class 12.

2. Key Concepts

2.1 Relations

A relation from set $A$ to set $B$ is a subset of the Cartesian product $A\times B$.

Example:
Let $A=\{1,2\},B=\{x,y\}$
A relation $R$ can be $R=\{(1,x),(2,y)\}$

Types of Relations:

1. Reflexive: $aRa$ for all $a\in A$
2. Symmetric: If $aRb⟹bRa$
3. Transitive: If $aRb$ and $bRc⟹aRc$
4. Equivalence Relation: Reflexive, symmetric, and transitive

2.2 Functions

A function $f:A\to B$ assigns each element of $A$ to exactly one element of $B$.

• Domain: Set $A$ (input)
• Codomain: Set $B$ (potential outputs)
• Range: Actual outputs in $B$ corresponding to elements of $A$

Types of Functions:

1. One-to-One (Injective): Each element of domain maps to a unique element in codomain.
2. Onto (Surjective): Every element of codomain is mapped by some element of domain.
3. Bijective: Both one-to-one and onto.
4. Constant function: Same output for every input.
5. Identity function: $f(x)=x$

3. NCERT Solutions – Step by Step

Example 1: Determining a Relation

Problem: Let $A=\{1,2,3\}$. Define a relation $R=\{(1,2),(2,3),(3,1)\}$. Is $R$ symmetric?

Solution:
Check if $(a,b)\in R⟹(b,a)\in R$

• (1,2) → (2,1)  not in $R$
• So, $R$ is not symmetric.

Example 2: Checking a Function

Problem: Determine if f(x)=x2f(x) = x^2f(x)=x2 from $\mathbb{R}\to \mathbb{R}$ is a function.

Solution:
Every real number $x$ has exactly one square x2x^2x2.
 Yes, it is a function.

Check for Injective/Surjective:

• Not injective: $f(2)=f(-2)=4$
• Not surjective over $\mathbb{R}$ (no negative outputs)

Example 3: Composition of Functions

Problem: Let $f(x)=x+1,g(x)=2x$. Find $(g\circ f)(x)$ and $(f\circ g)(x)$

Solution:

$$(g\circ f)(x)=g(f(x))=g(x+1)=2(x+1)=2x+2$$ $$(f\circ g)(x)=f(g(x))=f(2x)=2x+1$$

Example 4: Inverse Function

Problem: Find the inverse of $f(x)=3x+2$

Solution:

1. Set $y=3x+2$
2. Solve for $x$: x=y−23x = \frac{y-2}{3}x=3y−2​
3. Inverse: f−1(x)=x−23f^{-1}(x) = \frac{x-2}{3}f−1(x)=3x−2​