Piecewise Functions – Different Rules for Different Intervals
A piecewise function is defined by two or more different rules, each applying only over its own specific part of the domain. Instead of one formula covering every input, a piecewise function switches between formulas depending on where the input falls – graphically, this often produces a curve made of visibly different pieces stitched together at specific x-values.
To evaluate a piecewise function at a given x, the first step is always the same: check which interval x falls into, and only then apply that interval's rule. Using the wrong piece is the single most common mistake – always check the condition first.
Evaluating a Piecewise Function
Check which condition the input satisfies, then apply only that piece's rule. Every input belongs to exactly one piece.
Since 5 ≥ 2, use the second rule: f(5) = 3(5) − 2 = 15 − 2 = 13.
Real-Life Application
- Tax brackets: tax rates change at specific income thresholds, a classic piecewise function.
- Postage costs: shipping prices often jump at specific weight thresholds.
- Parking fees: hourly parking rates often change after a set number of hours.
Key Takeaways
- A piecewise function uses different rules over different parts of its domain.
- Always check which interval an input falls into before applying a rule.
- Every input belongs to exactly one piece, never more than one.
Practice: Piecewise Functions
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