Introduction to Modular Functions and Graphs

Summary of the Lecture: Introduction to Modular Function and Graphs

In today’s video lecture, we focused on the concept of modular functions. We began by defining the modular function and then proceeded to illustrate this concept with basic examples, building our understanding by plotting the corresponding graphs.

Important Points

Definition of Modular Function

• A modular function, denoted as f(x) = |x|, operates with the modulus of x.
  • For x ≥ 0: f(x) = x
  • For x < 0: f(x) = -x
  • This definition ensures all outputs (function values) are non-negative.

Characteristics of a Modular Function

• The modulus of a number refers to its non-negative value. E.g., |2| = 2 and |-2| = 2.

Graph of y = |x| (Function Identity)

• Basic Approach: Use a simple table of values to plot points.
  • For x = 0, y = 0 (origin point)
  • For x = 1, y = 1
• The graph is a V-shaped curve that does not pass into the negative y-axis, reflecting across the x-axis for negative x values.

Domain and Range of y = |x|

• Domain: All real numbers. You can use any real number as x.
• Range: 0, ∞). The function value starts from 0 and goes to infinity.

Next Function: f(x) = |x - 2|

• Like the previous example, we plot this function similarly using specific points derived from the function's equation.
  • At x = 0, y = |0 - 2| = -2 (intersecting the y-axis at -2)
  • Find where the function crosses the x-axis (y = 0) by setting x - 2 = 0, therefore x = 2.
  • Plot the points (0, -2) and (2, 0).

Graphing f(x) = |x - 2|

• The lower part of y, which would be negative, is reflected upward due to the modulus function, ensuring all y values are non-negative.
• The resulting graph is another V-shaped curve opening upwards, starting from y = 0.

Domain and Range of f(x) = |x - 2|

• Domain: All real numbers, since any real number can be substituted for x.
• Range: 0, ∞). The function value starts from 0 and increases to infinity, just as in the basic modular function.

Lecture Notes Review

• Always ensure to understand that modular functions modify all negative outputs to positive, altering typical linear graphs into V-shapes.
• Understanding modular functions involves recognizing the transformation made by the modulus operation on standard linear functions.
• Practice with more examples and incrementally explore more complex scenarios in the upcoming classes.

If you need further clarification on any concept discussed, feel free to revisit the video lecture, pursue the playlist recommended, or prepare specific questions for the next session. Make sure to attend upcoming lectures for more advanced exercises and continue to engage with the material actively!