Overview
This lecture explains piecewise functions, covering their definition, evaluation, graphing, and real-life applications with example problems.
Definition and Structure of Piecewise Functions
- A piecewise function is defined by two or more equations, each for specific intervals in the domain.
- Each piece applies to values of the input variable within a certain range.
Evaluating Piecewise Functions
- To find the value of a piecewise function at a given input, use the equation corresponding to the appropriate interval.
- Example: For ( f(x) = 3x + 2 ) if ( x \geq 0 ), ( f(0) = 2 ).
- Example: For ( f(x) = -x^2 + 3 ) if ( x < 0 ), ( f(-3) = -6 ).
Graphing Piecewise Functions
- Each interval is graphed according to its specific equation, with attention to interval boundaries.
- Example: ( f(x) = x + 3 ) for ( x \geq 0 ); ( f(x) = -x^2 + 3 ) for ( x < 0 ).
- Graphing tools like Desmos can be used to visualize piecewise functions.
More Examples and Applications
- Given ( f(x) = x + 2 ) if ( x \leq 2 ), and ( f(x) = -x + 3 ) if ( x \geq 2 ), ( f(-5) = -3 ) and ( f(3) = 0 ).
- Application 1: Mobile plan charging 300 pesos for up to 100 texts, and 1 peso per text after; modeled as a piecewise function.
- Application 2: Tricycle fare: 20 pesos for the first kilometer, then 5 pesos per additional 0.5 km; modeled by ( f(d) = 20 ) for ( d \leq 1 ), ( f(d) = 20 + 5 \frac{d-1}{0.5} ) for ( d > 1 ).
Key Terms & Definitions
- Piecewise Function — A function defined by multiple equations for different intervals of the domain.
- Domain — Set of all possible input values for a function.
- Interval — A range of values where a specific equation applies in the function.
- Quadratic Function — A function involving ( x^2 ), forming a parabola when graphed.
Action Items / Next Steps
- Practice evaluating and graphing piecewise functions for various intervals.
- Complete any assigned exercises involving real-life applications of piecewise functions.
- Use graphing tools like Desmos to sketch piecewise functions as homework.