This Warm-up challenges students to decode a short message, prompting them to think about what was done to produce the coded message so that it could be undone. The reasoning they do here paves the way for thinking about reversing the process that defines a function and about using outputs as inputs. This will get students ready to make sense of the inverse of a function.

It is not essential that students decode the message. What is important is the awareness that cracking the code involves a reversal process.

If one or more students were able to decode the message, ask them to share their result and how they went about decoding it. Otherwise, solicit some comments on the strategies they tried and any hypotheses on how the message was encoded. (Students are likely to hypothesize that the code is related to the position number of each letter in the alphabet.)

Then, reveal the original message. Give students a brief moment to think about how it was coded. Discuss questions such as:

• "Suppose you found out that each letter in the message was encoded by using the letter 3 places after the original letter. How would you decode the secret message?" (By using the letter 3 places before that coded letter.)
• "The code JRRG IRRG is produced with the same method. What does it say?" (GOOD FOOD) "What about EDEB?" (BABY)

Introduce Caesar shift cipher (or shift cipher) as a way to encrypt a message by shifting its alphabet position a certain number of places. The message in the Warm-up is called "a shift of 3" because it substitutes each letter in the original message with the letter 3 places after. A table could be used as a key. It enables us to easily see the plain-text alphabet and the cipher-text alphabet. Here is an example for a shift of 3.

A similar table could be used as a key for a shift of 5, 2, -3, or any other number.

Tell students that they will use the idea of writing and decoding a cypher to think about functions.

In this activity, students are introduced to inverse functions. They begin by creating their own shift cipher and using it to encode a message. After exchanging messages with a partner and decoding each other's messages, students describe the encoding and decoding process in terms of mathematical functions.

The encoding and decoding portion of the activity can be simplified by assigning a shift number to students.

The Task Statement includes a table students can use to create a key to help with encoding and decoding.

Another tool that can be used as a key for multiple shift cipher is a cipher wheel, as shown here. If desired and if time permits, consider constructing one or asking students to do so. A template and instructions are included in the blackline master.

In this activity, students continue to develop an understanding of inverse functions, using currency exchange as a context. Students convert values in one currency into another and then the other way around. They make sense of the former as a function and the latter as the inverse of the first function.

The reasoning here builds on students' prior work on solving equations for a variable. Earlier in the course, students had isolated a variable to highlight a quantity or make it easier to compute the values of the variable. What is new here is the idea of inverse function: the recognition that a variable that is an output in one function (the one that is isolated) is an input in the inverse function, and vice versa.

Some students may choose to graph the first function () and use the graph to find all the unknown values in either currency. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).