Overview
This lecture covers piecewise functions, including their definition, evaluation, graphing, and real-world applications with examples and sample problems.
Piecewise Functions: Definition and Evaluation
- A piecewise function is defined by two or more equations, each valid over specific intervals of the domain.
- To evaluate a piecewise function, determine which equation to use based on the value of x.
- Example: For ( f(x) = 3x+2 ) if ( x \geq 0 ), and ( f(x) = -x^2+3 ) if ( x < 0 ).
- To find ( f(0) ), use ( 3x+2 ): ( 3(0)+2 = 2 ).
- To find ( f(-3) ), use ( -x^2+3 ): (-(-3)^2+3 = -9+3 = -6 ).
Graphing Piecewise Functions
- Graph each part according to its formula and specified interval.
- For example, ( y = x+3 ) for ( x \geq 0 ) is a straight line starting at ( x=0 ).
- ( y = -x^2+3 ) for ( x < 0 ) is a downward-opening parabola left of ( x=0 ).
- Use graphing tools (like Desmos) to plot and visualize piecewise functions.
More Examples and Practice
- For ( f(x) = x+2 ) if ( x \leq 2 ), ( f(x) = -x+3 ) if ( x \geq 2 ): ( f(-5) = -3 ), ( f(3) = 0 ).
- Always substitute the x-value into the correct equation based on its interval.
Real-World Applications
- Mobile plan: Pay 300 pesos/month for up to 100 texts, then 1 peso per text after 100. Represent this with a piecewise function for cost as a function of messages sent.
- Tricycle fare: 20 pesos for first kilometer, plus 5 pesos per half kilometer after. Piecewise function handles fare calculation based on distance.
- Retail pricing, utilities, and similar scenarios can use piecewise functions to model changing rates.
Key Terms & Definitions
- Piecewise Function — A function defined by different formulas over different intervals of its domain.
- Interval — A range of input values (domain) where a specific formula applies.
- Graph — A visual representation that shows how each part of a piecewise function behaves.
Action Items / Next Steps
- Practice evaluating and graphing piecewise functions.
- Complete sample problems on real-world applications from the lecture.
- Try graphing piecewise functions using online tools such as Desmos.