Piecewise functions show up on every algebra 2 and precalculus exam — and most students freeze the moment they see a function defined by two or three different rules at once. Which branch do you use? What does that open circle mean? Does the graph actually connect? This guide cuts straight to those questions.

**TLDR: Piecewise Functions** walks you through every skill you need: reading the notation, picking the right branch when evaluating, sketching each piece over its subdomain with correct open and closed endpoints, and testing whether the function is continuous at the seams using one-sided limits. Along the way you'll see how absolute value, floor, ceiling, and sign functions are just piecewise definitions in disguise.

The final section grounds everything in the real world — tax brackets, tiered shipping costs, and phone data plans are all piecewise functions. Once you see that, the notation stops feeling arbitrary.

This guide is written for high school students in algebra 2, precalculus, or an introductory calculus course, and for college students who need a fast, no-filler refresher. Every term is defined in plain language the first time it appears. Worked examples are shown step by step. Common mistakes — like plugging an input into the wrong branch — are called out and corrected directly.

Short by design, stripped to essentials, and organized around exactly what students get wrong. If you want to walk into your next exam knowing how to handle piecewise functions confidently, grab this guide and get to work.

• Read piecewise notation and evaluate f(x) at any input by picking the right rule
• Graph piecewise functions, including open and closed endpoints at boundaries
• Recognize and use absolute value, step, and sign functions as piecewise functions
• Test continuity at boundary points by comparing one-sided values
• Build a piecewise function that models a real situation like tax brackets or shipping costs

1. 1. What a Piecewise Function Actually Is
   Introduce the idea of a function defined by different rules on different parts of its domain, with notation walk-through.
2. 2. Evaluating Piecewise Functions
   Step-by-step process for plugging in inputs: pick the right branch first, then compute.
3. 3. Graphing Piecewise Functions
   How to sketch each branch over its subdomain and handle open/closed circles at the seams.
4. 4. Special Cases: Absolute Value, Step, and Sign Functions
   Recast familiar functions as piecewise definitions and meet the floor, ceiling, and sign functions.
5. 5. Continuity at the Seams
   Test whether a piecewise function connects smoothly at boundary points using one-sided limits.
6. 6. Where Piecewise Functions Show Up
   Real-world models: tax brackets, shipping costs, phone plans, and why piecewise is the natural language for thresholds.