Lecture Notes: Functions in Mathematics

Introduction

• Welcome to another beginners course session focused on functions.
• Functions can be intimidating to many students, but with foundational understanding, they become manageable.
• The session is part of a 21-day challenge where foundational concepts are built for further mathematical concepts.

Key Topics Discussed

Ordered Pairs and Functions

• Introduction to ordered pairs and their role in functions.
• Understanding domain and range through examples and visual tests (e.g., vertical line test).
• Different types of functions: algebraic, linear, quadratic, etc.

Basic Operations with Functions

• Addition, subtraction, multiplication, and division of functions.
• Example calculations for better understanding.

Understanding Domain and Range

• Domain: Set of all acceptable inputs into the function.
• Range: Set of all possible outputs from the function.
• Two main rules:
  • Denominator of a fraction shouldn't be zero.
  • Expression inside a square root should be non-negative.

Visual Tests for Functions

• Vertical Line Test for determining if a graph represents a function.
• Explanation with examples to solidify understanding.

Relations and Mappings

• Explanation of mapping elements in set A to elements in set B.
• Discussion on uniqueness of mappings in functions.
• Examples using analogy of children (set A) and their mothers (set B).

Domain and Range in Graphs

• Graphical representations can simplify understanding of domain and range.
• Application of waving curve method for solving inequalities.

Special Function Types

• Linear, quadratic, cubic, and modulus functions briefly introduced.
• Exponential and logarithmic functions as inverses of each other.

Advanced Concepts and Application

• Techniques for breaking down complex function problems into simpler parts.
• Algebra of functions in problem-solving.
• Use of symbolic representations and transformations in understanding functions.

Conclusion

• Functions are a fundamental part of mathematics and are applicable across various chapters and topics.
• Regular practice and understanding of foundational rules will aid in solving complex problems.
• Continuous practice and engagement are crucial for mastering functions.

Resources and Tools

• Sessions and resources are available through the provided playlist and tiny URL links.
• Additional practice through PYQ sessions, NTA sessions, and DPPs shared in the Telegram group of Mathematically Inclined.