Piecewise Functions and the Basic Graphs

A piecewise function is simply a function made of pieces: one rule for some inputs, a different rule for others. Phone plans work like this, with one price up to your data limit and another price after. The absolute value up top is the classic example, and it has two small pieces: $x$ when $x \ge 0$ (leave positive numbers alone) and $-x$ when $x < 0$ (flip negative numbers to positive). So $|x|$ just means "distance from zero." The curly-bracket notation looks formal, but it's just a list of rules with a condition next to each.

Evaluating. Look at the conditions first, find the one your input satisfies, and use only that rule. The other rules simply don't apply there.

Graphing. Draw each piece only over its own interval. Where a piece stops, use a closed dot if that endpoint is included and an open dot if it isn't, so it's clear which rule owns the boundary.

The basic graphs. These five show up again and again. Later in this unit you'll shift, stretch and flip them to get almost everything else, so it's worth knowing their shapes and a couple of points on each.

A common mistake. With a tax schedule like the one pictured below, it's tempting to think that crossing into a higher bracket taxes your whole income at the higher rate. It doesn't. Each rate applies only to the dollars inside its own bracket, so the tax function has corners where the rate changes but no jumps.

Tax brackets, shipping rates, phone plans with overage charges and a thermostat's on/off behavior are all piecewise functions.