Functions as Models Lecture Notes

Introduction

• Continuation of last week's discussion about functions as models.
• Recap of five models:
  • Table of values
  • Mapping diagram
  • Equation
  • Graph
  • Machine
• Differences between relation and function.

Domain

• Domain: Set of values that the variable x can take.
• Represents possible values that satisfy given functions or equations.

Examples of Functions as Equations

1. Function 2x + 1 and x² - 2x + 2
   • All real numbers satisfy the functions (fractions, decimals, whole numbers, integers, natural numbers).
2. Circle Equation
   • Not classified as a function due to its nature.
3. Negative Numbers and Imaginary Numbers
   • Negative inputs can lead to imaginary numbers; undefined at x = 1.
4. Function Notations
   • Functions can be represented in the format of f(x), g(x), etc.

Functions as Representations of Real-Life Equations

• Functions involve four fundamental operations:
  • Addition
  • Subtraction
  • Multiplication
  • Division

Example 1: Cost of Meals

• Function c(x) = 40x (cost of buying x meals at 40 pesos each).
• Example: Buying 5 meals costs 200 pesos.

Example 2: Fencing Problem

• 100 meters of fencing to enclose a rectangular area next to a river.
• Area function: A = x * y (length * width).
• Perimeter formula: 100 = 2L + 2W.
  • Set L = x and W = y.
  • Rearranging gives: 100 = x + 2y.
  • Solve for y: y = (100 - x)/2.
  • Substitute y back into area function: A = x * ((100 - x)/2) = 50x - 0.5x².

Piecewise Functions

• Defined by different formulas for each interval.
• Example 1: Distance from home to school, then school to market.
  • Total fare based on intervals.
• Example 2: Mobile plan charging 300 pesos monthly for 100 free messages; excess messages cost 1 peso each.
  • Function based on number of messages sent.

Conclusion

• Discussed the domain for each function.
• Explored real-life applications of functions, including piecewise functions.