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Imagine a streaming service that costs $10 a month for up to 50 hours of video, but charges $0.50 for every extra hour after that. A single algebraic equation cannot describe your bill—you need two different formulas depending on your watch time.

Anatomy of a Piecewise Function

A piecewise-defined function is a single function composed of multiple sub-functions, where each sub-function applies to a specific interval of the input called a sub-domain.
In 1837, mathematician Peter Gustav Lejeune Dirichlet revolutionized mathematics by defining a function not as a single algebraic formula, but as any rule pairing each input with exactly one output—paving the way for piecewise definitions.
How do we know which rule to calculate when given an input number?

The Two-Step Evaluation Method

Evaluating a piecewise function is an if-then test. You never plug an input into all the equations—only into the single matching rule.
First, perform Domain Matching by testing the input value against each conditional inequality to determine which specific sub-domain it satisfies.
Second, perform Substitution by evaluating only the corresponding algebraic expression for that chosen sub-domain.
What happens when your input sits directly on the boundary between two rules?

Boundary Values and Strict Inequalities