Piecewise Functions Math Activities

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Definition

A piecewise function is a function defined by different expressions or rules for different intervals of the input variable. Instead of one single equation, a piecewise function uses multiple equations, each applying to a specific part of the domain.

Example:

Summary

1. Definition and Structure

Each “piece” of the function has:
- Its own equation (expression)
- A condition (domain restriction) that determines when that equation is used.
The pieces are usually separated using braces {⋯ }.

2. Evaluating Piecewise Functions

Identify which part of the domain the input belongs to.
Use the correct expression for that interval.
Plug in the value and simplify.

3. Graphing Piecewise Functions

Plot each piece only on the interval where it’s defined.
Include open or closed dots depending on whether the interval includes the endpoint.
Graphs can look disconnected or have sharp changes (“jumps”).

4. Applications

Real-world situations with different conditions or rates:
- Shipping costs that vary by weight
- Tax brackets
- Speed limits over time or zones
- Pay based on hours worked (e.g., overtime)

5. Common Challenges

Correctly interpreting which piece to use.
Dealing with endpoints (using open vs. closed circles).
Ensuring continuity or identifying discontinuity in the grap