Overview
This lesson covers how to evaluate piecewise functions by determining which expression to use based on the value of x and calculating specific function values.
Introduction to Piecewise Functions
- A piecewise function is defined by different expressions depending on the value of x.
- Example: f(x) = 4x + 5 when x < 2, and f(x) = 3x - 8 when x ≥ 2.
Evaluating Piecewise Functions: Example 1
- For f(-2): Use 4x + 5 because -2 < 2; f(-2) = 4(-2) + 5 = -8 + 5 = -3.
- For f(2): Use 3x - 8 because 2 ≥ 2; f(2) = 3(2) - 8 = 6 - 8 = -2.
- For f(5): Use 3x - 8 because 5 > 2; f(5) = 3(5) - 8 = 15 - 8 = 7.
Evaluating Piecewise Functions: Example 2
- f(x) = x² + 3x - 7 when x < -1
- f(x) = 5x + 6 when -1 ≤ x < 2
- f(x) = 12 when x = 2
- f(x) = x³ + 4 when x > 2
- For f(-4): Use x² + 3x - 7; f(-4) = (-4)² + 3(-4) - 7 = 16 - 12 - 7 = -3.
- For f(0): Use 5x + 6; f(0) = 5(0) + 6 = 0 + 6 = 6.
- For f(2): Use 12; f(2) = 12.
- For f(3): Use x³ + 4; f(3) = 3³ + 4 = 27 + 4 = 31.
Key Terms & Definitions
- Piecewise Function — A function defined by different expressions for different intervals of the input variable x.
Action Items / Next Steps
- Practice evaluating piecewise functions for various x values.
- Review your class assignment or textbook for additional exercises on this topic.