Piecewise Models and Systems

A piecewise function uses different rules on different intervals. Tax brackets and tiered pricing are the everyday examples.

Reading the definition

Each rule comes with the interval it applies to. Evaluating means choosing the right piece first.

Break points

The interval boundaries are where the rule changes. Each boundary belongs to exactly one piece.

Continuous or not

If both pieces agree at a boundary the graph joins smoothly. If they disagree there is a jump.

Comparing two options

Two pricing plans are two functions. Setting them equal finds the usage where they cost the same.

Answering the question

The intersection is the break-even point. Which plan wins depends on which side of it you are on.

When one rule does not cover everything

A piecewise function uses different formulas on different intervals. Tax brackets, postage rates and tariffs all work this way, so the form is far more common in practice than in textbooks.

The boundary decides continuity

The pieces agree at a boundary only if their formulas give the same value there. Finding a constant that makes them agree is the standard problem, and a jump is a real feature rather than an error.

Choose the piece before computing

To evaluate at a point, first identify which interval contains it. Using the wrong piece is the standard error and it is entirely avoided by checking the interval first.

Systems model competing constraints

Two conditions holding at once become a system; several become a feasible region. Break-even analysis, resource allocation and scheduling are all systems problems dressed in context.