# Lesson 12

Piecewise Functions

### Lesson Narrative

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).

### Learning Goals

Teacher Facing

• Interpret a graph of a piecewise function or the rules given in function notation, and explain the rules (orally and in writing) in terms of a situation.
• Sketch a graph that represents the rules of a piecewise function, paying special attention to the endpoints of each interval.
• Understand a piecewise function as a function defined by different rules for different intervals of the domain.

### Student Facing

• Let’s look at functions that are defined in pieces.

### Student Facing

• I can make sense of a graph of a piecewise function in terms of a situation, and sketch a graph of the function when the rules are given.
• I can make sense of the rules of a piecewise function when they are written in function notation and explain what they mean in the situation represented.
• I understand what makes a function a piecewise function.

Building Towards

### Glossary Entries

• piecewise function

A piecewise function is a function defined using different expressions for different intervals in its domain.