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."

Related Topics

Absolute Value FunctionsWhat is a Function?Function Transformations

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.