What is Relations and Functions Class 12: Complete NCERT Guide

Definition and Basics of Relations in Class 12 Mathematics

In Class 12 NCERT Mathematics, a relation between two sets $A$ and $B$ is a subset of the Cartesian product $A \times B$. It consists of ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. This means each element of $A$ is related to one or more elements of $B$.

Key points:

• A relation can be represented as a set of ordered pairs.
• It can also be shown using a mapping diagram or a graph.
• Relations are not necessarily functions because an element in $A$ can relate to multiple elements in $B$.

Example: If $A = \{1, 2\}$ and $B = \{x, y\}$, a relation $R$ could be $\{(1, x), (2, y), (2, x)\}$.

This chapter lays the foundation for understanding how elements from one set correspond to elements in another, a critical concept for further study.

Understanding Functions: Special Relations in Class 12 NCERT

A function is a special type of relation where every element of the domain $A$ is related to exactly one element of the codomain $B$. In other words, no element in $A$ maps to more than one element in $B$.

Formal definition:

A function $f$ from set $A$ to set $B$ is denoted as $f: A \to B$ such that for each $a \in A$, there exists a unique $b \in B$ with $f(a) = b$.

Important terms:

• Domain: The set $A$ from which inputs are taken.
• Codomain: The set $B$ where outputs lie.
• Range: The actual set of outputs $\{f(a) : a \in A\}$.

Example: If $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$, then for each real number $x$, there is a unique output $x^2$.

Functions are fundamental in Class 12 Maths as they model real-world relationships and are used in calculus, algebra, and more.

Types of Functions Explained for Class 12 Students

Class 12 NCERT Maths classifies functions into several types based on their properties:

Knowing these types helps in solving problems related to inverse functions and composition, which are important for exams.

Worked example: Check if $f(x) = 3x + 2$ is one-one.

If $f(a) = f(b)$, then $3a + 2 = 3b + 2 \Rightarrow 3a = 3b \Rightarrow a = b$. So, $f$ is one-one.

Domain, Range, and Co-domain: Key Concepts in Relations and Functions

Understanding domain, range, and codomain is crucial for mastering relations and functions.

• Domain: The set of all possible inputs.
• Codomain: The set into which all outputs are constrained.
• Range: The set of actual outputs produced by the function.

For example, consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \sqrt{x}$.

• Domain: $[0, \infty)$ (since square root of negative numbers is not real)
• Codomain: $\mathbb{R}$ (all real numbers)
• Range: $[0, \infty)$ (outputs are non-negative)

This distinction is important for solving problems and understanding the behaviour of functions in Class 12.

Composition and Inverse of Functions in Class 12 NCERT Maths

Two important operations on functions are composition and inverse.

Composition of Functions

If $f: A \to B$ and $g: B \to C$ are functions, then the composition $g \circ f$ is a function from $A$ to $C$ defined by:

$$ (g \circ f)(x) = g(f(x)) $$

Example: If $f(x) = 2x$ and $g(x) = x + 3$, then:

$$(g \circ f)(x) = g(2x) = 2x + 3$$

Inverse of a Function

A function $f: A \to B$ has an inverse $f^{-1}: B \to A$ if $f$ is bijective. The inverse satisfies:

$$ f^{-1}(f(x)) = x, \quad \text{for all } x \in A $$

Example: If $f(x) = 3x + 1$, then $f^{-1}(y) = \frac{y - 1}{3}$.

Mastering these concepts is essential for Class 12 students to solve advanced problems efficiently.

Relation vs Function: A Clear Comparison for Class 12 Students

Understanding the difference between a relation and a function is fundamental.

This comparison helps students avoid common mistakes in exams and understand the chapter deeply.