ACT Preparation / Unit 10 ACT 0911 worked example, 60 to 90 seconds
Evaluating a Piecewise Function
The idea
A piecewise function is evaluated in three steps. First, read each piece's interval, noting whether each endpoint is included. Second, locate the input in exactly one interval. Third, apply only that piece's formula. On a graph, an open circle marks an excluded endpoint and a closed circle an included one.
In practice
Worked example: a parking garage charges $5 for up to 2 hours, $5 plus $3 for each extra hour up to 6 hours, and a flat $20 above 6 hours. As a function of hours h: P(h) = 5 for 0 < h ≤ 2; 5 + 3(h − 2) for 2 < h ≤ 6; and 20 for h > 6. Then P(2) = 5, P(5) = 5 + 9 = 14, and P(7) = 20.
Where people slip
An input at a boundary belongs to the piece whose interval includes it. P(6) uses the middle piece, 5 + 12 = 17, not the flat 20.
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