Definition
Piecewise functions are functions defined by different algebraic rules (sub-functions) for different, non-overlapping intervals of their domain.
Why It Matters
One rule doesn’t always fit all. Piecewise functions allow us to describe systems that change their behavior abruptly, modeling real-world “jumps” like tax brackets, physics phase changes, and step-based pricing.
Core Concepts
- Piecewise Mechanics: A piecewise function uses a bracket to list various expressions alongside their corresponding domain constraints.
- How to read: “The piecewise function f of x.”
- Meaning: The rule depends on the specific interval of .
- Evaluation: To evaluate, you must first identify which interval the input falls into, then use only that specific rule to calculate the output.
- How to read: “Evaluate f at x.”
- Meaning: Substitute the value of into the correct sub-function.
- Continuity: The “seams” where rules switch are critical points; limits must be evaluated from both sides to ensure the graph doesn’t break or jump.