Math101 · Independent Ontario learning library
Math101Piecewise Functions
A piecewise function uses different formulas on different parts of its domain. The input condition selects exactly which rule to apply.
Piecewise functions model tax brackets, rates, shipping costs, control systems, and any process whose rule changes across thresholds.
Intuition and core definition
A piecewise function uses different formulas on different parts of its domain. The input condition selects exactly which rule to apply. Endpoint symbols prevent ambiguity: a well-defined function cannot assign two different outputs to the same included input.
Notation, language, and conditions
$f(x)=\begin{cases}x+2,&x<1\\x^2,&x\ge1\end{cases}$ pairs each formula with a condition. Open and closed endpoints on a graph correspond to strict and inclusive inequalities. The formula’s own restrictions also apply within its stated interval.
Why this idea matters
A piecewise function assigns formulas to stated input regions, so boundary symbols are as important as the formulas themselves.
A dependable method
- Locate the requested input on the condition intervals.
- Select only the rule whose condition includes that input.
- Substitute and evaluate using that rule.
- For a graph, draw each formula only over its assigned interval and mark endpoints open or closed.
- Check that domain pieces cover intended inputs without conflicting overlaps.
