### Notes from Toppers

**Detailed Notes on Relations and Functions**

**1. Types of Relations**
**NCERT Reference**: Chapter 1, Relations and Functions, Class 11

**One-to-one**: A relation R from a set A to a set B is said to be one-to-one if every element of B has a unique pre-image in A.**One-to-many**: A relation R from a set A to a set B is said to be one-to-many if every element of B has at least one pre-image in A.**Many-to-one**: A relation R from a set A to a set B is said to be many-to-one if every element of B has more than one pre-image in A.**Many-to-many**: A relation R from a set A to a set B is said to be many-to-many if every element of B has more than one pre-image in A.

**2. Functions**
**NCERT Reference**: Chapter 1, Relations and Functions, Class 11

**Definition of a Function**: A function f from a set A to a set B is a relation R from A to B that assigns to each element x in A a unique element y in B, denoted by f(x).**Injective**: A function f from a set A to a set B is said to be injective (one-to-one) if for any two distinct elements a1 and a2 in A, f(a1) ≠ f(a2).**Surjective**: A function f from a set A to a set B is said to be surjective (onto) if for every element b in B, there exists at least one element a in A such that f(a) = b.**Bijective**: A function that is both injective and surjective is called a bijective function.

**3. Algebra of Functions**
**NCERT Reference**: Chapter 1, Relations and Functions, Class 11

**Addition**: (f + g)(x) = f(x) + g(x)**Subtraction**: (f - g)(x) = f(x) - g(x)**Multiplication**: (f * g)(x) = f(x) * g(x)**Division:**(f / g)(x) = f(x) / g(x) (g(x) ≠ 0)

**4. Graphical Representation of Functions**
** NCERT Reference**: Chapter 2, Functions, Class 12

- Plotting graphs allows for the visualization of the behavior of functions.
- Graphs can provide information such as the domain, range, and key features (maxima, minima, asymptotes).

**5. Limits and Continuity**
** NCERT Reference**: Chapter 2, Limits and Derivatives, Class 11

**Limits**:- Limit of a function f(x) as x approaches a (denoted by lim_(x->a) f(x)) represents the value “L” if for any positive number ε, there exists a positive number δ such that if 0 < |x − a| < δ, then |f(x) − L| < ε.
- Types of discontinuities include removable, jump, and infinite discontinuities.
- Limit laws and theorems simplify limit calculations.

**6. Applications of Functions**
** NCERT Reference**: Chapter 2, Functions, Class 12

**Optimization Problems:**Functions can be used to optimize a quantity, such as finding maximum profit, maximum volume, or minimum cost.**Curve Sketching**: Analyzing the behavior of functions based on their graphs aids in understanding their characteristics and properties.

**7. Sequences and Series**
** NCERT Reference**: Chapter 2, Sequences and Series, Class 11

**Sequences**: A sequence is an ordered list of numbers {a1, a2, a3, …}.**Limits of sequences**: If for any given ε > 0, there exists a positive integer M such that |am - L| < ε whenever m > M, then L is the limit of the sequence as m approaches infinity (denoted as lim_(m->∞) am = L).**Convergence and Divergence**: A series converges if the sum of its terms approaches a finite limit, and diverges if it approaches infinity or does not have a finite limit.