Lecture Notes on Magnet Brains and Modulus Function

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Today's Topic: Modulus Function

• Presenter: Zeeshan, Maths Mentor
• Module Focus: Understanding the 'King' function or modulus function.

Understanding Modulus Function

Basic Concepts

• Square Root Example:
  • For example: √4 = 2
  • If x² = 4, x = ±2.
• Modulus Function (|x|):
  • |x| = x if x ≥ 0
  • |x| = -x if x < 0
• Example Discussion:
  • |2| = 2
  • |-2| = 2
  • |0| = 0

Properties of Modulus Function

• Graph: V-shaped graph, symmetric about the y-axis.
  • Positive inputs and their corresponding outputs lie in the first quadrant.
  • Negative inputs and their corresponding outputs lie in the second quadrant.
• Magnitude Representation:
  • Absolute value shows the magnitude of a number irrespective of its sign.
  • It represents the distance from origin to the number on the number line.
• Alternate Name:
  • Also known as the absolute value function.

Calculation using Modulus Function

Formula

• |x| = x if x ≥ 0
• |x| = -x if x < 0

Examples

• |2| = 2
• |-4| = 4
• |-1.5| = 1.5
• |-√3| = √3

Domain and Range of Modulus Function

Domain

• Definition: Set of all possible input values (x-values)
• For |x|: All real numbers (R)
  • R = {x | x ∈ R}

Range

• Definition: Set of all possible output values (y-values)
• For |x|: Non-negative real numbers [0, ∞)

Modulus Function in Mixed Functions

• Example: f(x) = a|x| + k
• Domain: All real numbers (R)
• Range: Depends on the values of a and k. Cannot be generalized as easily as a pure modulus function.

Summary

• Magnet Brains: Free, high-quality education for students from Kindergarten to Class 12, available in English and Hindi mediums.
• Modulus Function: A fundamental 'King' function in math, with properties that include absolute value calculations and graph representations.
• Domain and Range: Comprehensive coverage of what values can be input and what the resultant output values could be.

Closing

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