Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions, showing key features by hand in simple cases and with technology when the functions are more complicated.
Classify step versus piecewise-linear behavior from evidence
Problem
Classify the graph for this situation as step or piecewise-linear: Shipping costs \(\$4\) for weights \(0<w\le~1~\text{lb}\) and \(\$7\) for \(1<w\le~3~\text{lb}\).
Big Picture
What this problem is really about
A function family is determined by how the output behaves within each interval, not merely by the number of pieces. We’ll check whether each band is constant or changing, look for an abrupt jump, and use the threshold inequalities to assign boundary ownership.
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