Overview
This lecture introduces piecewise functions, explains how to evaluate and graph them, and provides real-world applications using mathematical examples.
Piecewise Functions: Definition and Structure
- A piecewise function is defined by multiple equations, each applying to a specific interval of the domain.
- Each "piece" of the function has its own rule depending on the value of the variable.
Evaluating Piecewise Functions
- To evaluate a piecewise function, determine which interval the input belongs to and use the corresponding formula.
- Example: If ( f(x) = 3x + 2 ) for ( x \geq 0 ) and ( f(x) = -x^2 + 3 ) for ( x < 0 ), then ( f(0) = 2 ) and ( f(-3) = -6 ).
- Choose the correct formula based on whether the input is positive, zero, or negative.
Graphing Piecewise Functions
- When graphing, plot each formula only over its defined interval.
- Linear functions appear as straight lines; quadratic functions appear as curves, restricted to their intervals.
- Use tools like Desmos or graph paper to visualize.
More Examples
- For ( f(x) = x + 3 ) if ( x \geq 0 ), ( f(x) = -x^2 + 3 ) if ( x < 0 ): ( f(-5) = -22 ), ( f(3) = 6 ).
- For ( f(x) = x + 2 ) if ( x \leq 2 ), ( f(x) = -x + 3 ) if ( x \geq 2 ): ( f(-5) = -3 ), ( f(3) = 0 ).
Real-World Applications of Piecewise Functions
- Mobile plan: Charges 300 pesos for up to 100 texts; each additional text costs 1 peso more. Model as a piecewise function.
- Tricycle fare: 20 pesos for the first kilometer, then 5 pesos per additional 0.5 km. Use a piecewise function to calculate total fare.
Key Terms & Definitions
- Piecewise Function — A function defined by different expressions based on the input interval.
- Domain — The set of input values for a function.
- Interval — A range of input values where a particular formula applies.
Action Items / Next Steps
- Practice evaluating and graphing piecewise functions with given examples.
- Use graphing tools like Desmos for visualization exercises.
- Solve the mobile plan and tricycle fare problems using piecewise functions.