What is Relations and Functions Class 11: Definition & Concepts

Definition of Relations in Class 11 Mathematics

In Class 11 NCERT Maths, a relation between two sets $A$ and $B$ is defined as a subset of the Cartesian product $A \times B$. It consists of ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

• If $A = \{1, 2\}$ and $B = \{x, y\}$, then $A \times B = \{(1,x), (1,y), (2,x), (2,y)\}$.
• A relation $R$ could be $\{(1,x), (2,y)\}$, which is a subset of $A \times B$.

Relations help us understand how elements from one set are connected to elements of another. They are foundational for defining functions.

Key points:

• Relations can be represented as sets of ordered pairs.
• They can also be shown using matrices or graphs.

Example: If $A = \{1, 2, 3\}$ and $B = \{4, 5\}$, a relation $R$ might be $\{(1,4), (2,5)\}$.

Understanding Functions: The Special Type of Relation

A function is a special type of relation from set $A$ (domain) to set $B$ (codomain) where every element in $A$ is related to exactly one element in $B$.

• Formally, $f: A \to B$ is a function if for every $a \in A$, there exists a unique $b \in B$ such that $(a, b) \in f$.

Important terms:

• Domain: Set of all inputs ($A$).
• Range: Set of all outputs actually mapped ($\subseteq B$).

Example: If $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$, then $f = \{(1,4), (2,5), (3,6)\}$ is a function.

Non-function example: Relation $R = \{(1,4), (1,5)\}$ is not a function because $1$ maps to two outputs.

Functions are essential in mathematics because they describe deterministic relationships.

Types of Functions Explained for Class 11 Students

Functions can be classified based on how elements in the domain and codomain relate:

One-One Function: No two inputs share the same output.

Onto Function: All elements in the codomain are covered.

Bijective Function: Perfect pairing between domain and codomain.

Understanding these helps in solving problems on inverse functions and composition.

Domain, Range, and Co-domain: Key Concepts

When studying functions, it is crucial to distinguish these terms:

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

Example: Consider $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$.

• Domain = $\mathbb{R}$ (all real numbers).
• Co-domain = $\mathbb{R}$.
• Range = $[0, \infty)$ (since squares are never negative).

Always remember: range $\subseteq$ co-domain.

This distinction is important in understanding function behaviour and solving related NCERT problems.

Composition and Inverse of Functions: How They Work

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 $g \circ f: A \to C$ is defined as:

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

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

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

Inverse of a Function

A function $f$ has an inverse $f^{-1}$ if it is bijective. The inverse reverses the mapping:

$$ f^{-1}(f(x)) = x $$

Example: If $f(x) = x + 5$, then

$$ f^{-1}(y) = y - 5 $$

Understanding these concepts is vital for Class 11 students to solve advanced problems.

Relation vs Function: A Quick Comparison

It's important to distinguish between relations and functions. Here's a comparison table:

Remember: All functions are relations, but not all relations are functions.