Lecture Notes: Function Notation

Introduction to Function Notation

• Importance of Slides:
  • Initial slides are crucial for understanding the concept.
  • Focus on definitions and meanings without the instructor's narration.

Traditional vs. Function Notation

• Traditional Method:
  • Given: y = x^2 + 1
  • Find y when x = 2:
    • Solution: y = 5
• Function Notation:
  • Given: f(x) = x^2 + 1
  • Find f(2):
    • Solution: f(2) = 5
• Transition: Function notation will be used in grades 11 and 12.

Example Problems

Example 1

• Function: g(x) = 2x^2 - 3x + 1
• Part A: Find g(-2)
  • Calculation: g(-2) = 15
• Part B: Find g(4) - g(2)
  • Calculation: g(4) - g(2) = 18
• Part C: Find 3 * g(-5)
  • Calculation: 3 * g(-5) = 198
• Part D: Find g(t)
  • Solution: g(t) = 2t^2 - 3t + 1
• Part E: Find g(t + 5)
  • Solution: 2t^2 + 17t + 11
• Part F: Find g(1) - 3
  • Solution: -3
• Part G: Solve for x if g(x) = 21
  • Solutions: x = -5/2 or x = 4

Graphical Representation

• Using Graphs with Function Notation:
  • F(1): y ≈ 5.5
  • F(0): y = 7
  • Find X if F(X) = 0: x = 10
  • Find X if F(X) = 3: x ≈ 2.75
  • F(12): Does not exist (graph only shown up to x = 10)

Additional Problem

• Function: F(X) = X^2
  • Solution: x = 2 is the only value that satisfies F(X) = X^2

Example 3

• Given: F(X) is a linear function
• Points: (3, 4) and (0, -2)
• Equation: Y = 2X - 2

Advanced Problems

Example: G(X + 2)

• Function: g(x) = √(x + 4)
• Solution: √(x + 6)

Example: F(H(X))

• Given: F(X) = X^2 - X
• H(X): 4X - 12
• Simplification:
  • Result: 16X^2 - 100X + 156

Example: 2 * F(X) - H(X)*

• Simplification: 2X^2 - 6X + 12

Example: H(X + 1) - H(X) / 2

• Calculation:
  • Simplified Result: 2

Example: F(X) = G(-X^2)

• Solve for X
  • Solutions: x = ±2

Conclusion

• Importance of Understanding Function Notation:
  • Know definitions and purposes.
  • Review introductory slides meticulously.

Homework

• Listed on notes and course outline.