# Lesson 15

Inverse Functions

### Lesson Narrative

This lesson introduces students to inverse functions. The idea is developed through several contextual problems that each requires reversing a process and using outputs as inputs.

In the warm-up and the first activity, students encounter the idea of inverse functions as they use Caesar shift ciphers to encode and decode messages. In such ciphers, encoding involves shifting the position of a letter in the alphabet a certain distance and in a certain direction. Decoding then means undoing or reversing the action.

In the second activity, students solve problems about currency exchange. They convert an amount in one currency (dollars) to another (pesos), and work with inverse functions as they convert the other way around (from pesos to dollars).

Throughout the lesson, students reason repeatedly with linear functions, working forward and backward as they perform calculations with numerical values. They practice finding regularity through repeated reasoning, which they then apply to write expressions or equations to describe inverse functions (MP8). As in previous lessons in the unit, students continue to use multiple representations of functions and to make connections between them. In doing so, they make sense of quantities and relationships in concrete and abstract terms (MP2).

### Learning Goals

Teacher Facing

• Given a linear function in context, describe (using words or equations) its inverse function.
• Recognize that if one function takes $a$ as its input and gives $b$ as its output, its inverse function takes $b$ as its input and gives $a$ as the output.
• Understand that the inverse of a linear function can be found by reversing the process that defines the initial function.

### Student Facing

Let’s define functions forward and backward.

### Required Preparation

If use of a cipher wheel is desired for the Caesar Says Shift activity, prepare copies of the blackline master for that activity.

### Student Facing

• I understand the meaning of “inverse function” and how it could be found.
• When given a linear function that represents a situation, I can use words and equations to describe the inverse function.

Building On

Building Towards

### Glossary Entries

• inverse (function)

Two functions are inverses to each other if their input-output pairs are reversed, so that if one function takes $$a$$ as input and gives $$b$$ as an output, then the other function takes $$b$$ as an input and gives $$a$$ as an output.

You can sometimes find an inverse function by reversing the processes that define the first function in order to define the second function.

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