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.