Functions
AI Assistant

Piecewise Functions

Different rules for different parts of the domain

A piecewise function uses different formulas on different intervals. The graph on the right shows one rule for x < 0 (a parabola) and another for x ≥ 0 (a straight line). Real life is full of piecewise functions: tax brackets, shipping rates, phone plans.

The key questions are: Where do the pieces connect? Is the function continuous (no gap) at the boundary? What's the domain of each piece?

Try asking the AI "Is this function continuous at x = 0?" or "Build a tax bracket function."

Graph

FAQ

What is a piecewise function?
A piecewise function has different formulas on different intervals. For example, f(x) = x² when x < 0 and f(x) = 2x when x ≥ 0. Each "piece" has its own rule and its own domain.
How do I know if a piecewise function is continuous?
Check the boundary points. At x = 0 in our example: the left piece gives 0² = 0 and the right piece gives 2(0) = 0. Since both give the same value, the function is continuous there — no gap or jump.
What are some real-world piecewise functions?
Tax brackets (different rates for different income ranges), shipping costs (flat rate up to a weight, then per-pound), phone plans (included minutes, then per-minute charges), parking fees (first hour free, then hourly).
How do I graph a piecewise function?
Graph each piece on its own interval. Use a filled dot at endpoints that are included (≥ or ≤) and an open dot at excluded endpoints (> or <). Then connect the pieces.

Related Topics

Absolute Value FunctionsWhat is a Function?Function Transformations