Lesson 12

This lesson introduces a new type of function—piecewise-defined functions. A piecewise function uses multiple descriptions to define the function on different parts of the domain. All piecewise functions in this lesson are presented in the context of situations where different rules apply for different input values, such as different bike rental prices for different rental durations, or different postage rates for different letter weights.

In the bike rental activity, students are given the rules of a function and asked to graph the function. To do so, they pay attention to the boundary points, where one rule ends and another begins, and learn to use open and solid circles to indicate the correct output values for the input values that demarcate the rules.

Students are also introduced to the cases notation for piecewise functions. They practice interpreting it, which involves making use of the structure in the graph and in cases notation (MP7).

Most piecewise functions presented in the main activities are step functions, that is, the function values are constant on different parts of the domain and the graph shows different horizontal line segments. (In the optional activity, students graph piecewise functions that include non-constant segments.) In upcoming lessons, students will encounter absolute value functions, which are a particular type of piecewise function.

Advanced preparation, which includes data collection and organization, is needed for the next lesson. See Required Preparation for Absolute Value Functions (Part 1).