Choose the correct branch at a boundary

Try

A fictional community printer charges 10 cents per page for the first 20 pages and 6 cents for each additional page. It accepts whole jobs from 1 through 100 pages. Calculate the charges for 19, 20 and 21 pages. The last answer is 206 cents, not 126 cents: the cheaper rate applies only to the extra page.

Before writing a rule, identify the domain and the threshold. A pricing phrase can describe either marginal pricing or a discount on the whole job. These produce different functions, so the intended policy must be stated.

Predict

Predict whether the printer charge jumps down at 21 pages. Then consider a different policy charging 6 cents for every page once a job exceeds 20 pages. That policy would charge 126 cents for 21 pages, a sharp decrease from 200 cents. A discontinuity can represent a real policy rather than an algebra mistake.

Experiment

For the original marginal-price policy, write C(n)=10n for 1≤n≤20 and C(n)=200+6(n−20) for 20<n≤100, where C is cents and n is a whole-number page count.

Pages n	Applicable formula	Charge in cents
19	10n	190
20	10n	200
21	200+6(n−20)	206
30	200+6(n−20)	260

Plot individual points for the integer domain. If you also draw connecting line segments, label them as a continuous extension for visualization; fractional pages are not actual permitted jobs. This distinction prevents a helpful drawing from changing the model.

Understand

A piecewise function uses different formulas under different input conditions. Evaluate the condition before evaluating a formula. For f(x)=x² when x is negative and f(x)=2x+1 otherwise, the input −3 gives 9, not −5. At zero the second branch gives 1.

Conditions must cover the intended domain without assigning incompatible outputs. Using x<2 for one branch and x>2 for another leaves 2 undefined. That may be intended, but it must be visible. Using x≤2 and x≥2 overlaps at 2. An overlap is harmless only if both formulas give the same value there.

Endpoint markers encode this information on a graph. An open circle excludes a point from that branch; a filled point includes it. For f(x)=x+1 when x<0 and f(x)=x+4 when x≥0, draw an open circle at (0,1) and a filled circle at (0,4). The left and right nearby heights differ, so changing only the value at zero cannot remove the jump.

Matching values at a join is an algebra problem. If f(x)=mx−1 for x<3 and f(x)=x+2 for x≥3, the right branch value is 5. Matching requires 3m−1=5, hence m=2. The joined graph is continuous there, although the slopes of its two lines differ. Matching values and matching rates are separate conditions.

Apply

The delivery practice uses proportional billing rather than rounding each extra kilogram upward. Do not silently change that assumption. In a written policy model, give formulas, branch conditions, units and the full domain. Check one input below the threshold, the threshold itself and one above it.

For the conflicting-branches question, identify an input receiving two different values. Merely saying “the branches overlap” is insufficient: equivalent formulas can overlap without violating function uniqueness.

Continue

Composition passes an output through a second rule. When a function is piecewise, an intermediate output can determine which outer branch applies. Tracking that intermediate quantity becomes essential.