Lesson 15

Inverse Functions

15.1: What Does It Say? (10 minutes)


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.


Some students may be unfamiliar with the idea of ciphers or coded messages. Offer a brief introduction, if needed.

Give students a moment of quiet think time. If students struggle to get started after some time, consider giving a clue or two:

  • The letters A and I are the only two letters that can stand on their own (each letter can be a word).
  • The word “is” is the most common two-letter word in the English language.

Leave time for class discussion, even if students have not yet managed to decipher the code at that time.

Student Facing

Here is an encoded message, a message that has been converted into a code.


Can you figure out what it says in English? How was the original message encoded?

Student Response

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Activity Synthesis

If one or more students were able to decode the message, ask them to share their finding 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 down the alphabet, or 3 places after the original letter. How would you decode the secret message?" (By using the letter 3 places up the alphabet, or 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 down. 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.

plain text A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
cipher text D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

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.

15.2: Caesar Says Shift (15 minutes)


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 activity 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.

letters of alphabet written twice on segments inside of circle  


Arrange students in groups of 2. Tell partners that they are each to use a shift cipher to write a short secret message, exchange it with their partner's, and try to decode each other's secret message.

Student Facing

  1. Now it’s your turn to write a secret code!

    1. Write a short and friendly message with 3–4 words.
    2. Pick a number from 1 to 10. Then, encode your message by shifting each letter that many steps forward or backward in the alphabet, wrapping around from Z to A as needed.

      Consider using this table to create a key for your cipher.

      plain text A B C D E F G H  I   J  K L M N O P Q R S T U V W X Y Z
      cipher text                                                    
    3. Give your encoded message to a partner to decode. If requested, give the number you used.
    4. Decode the message from your partner. Ask for their number, if needed.
  2. Suppose \(m\) and \(c\) each represent the position number of a letter in the alphabet, but \(m\) represents the letters in the original message and \(c\) the letters in your secret code.

    1. Complete the table.

      letter in message        
      \(m\) 6 9 19 8
      letter in code        
    2. Use \(m\) and \(c\) to write an equation that can be used to encode an original message into your secret code.
    3. Use \(m\) and \(c\) write an equation that can be used to decode your secret code into the original message.

Student Response

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Student Facing

Are you ready for more?

There are 26 letters in the alphabet, so only the numbers 1–26 make sense for \(m\) and \(c\).

  1. Try using the equation that you wrote to encode the letters A, B, Y, and Z. Did you end up with position numbers or \(c\) values that are less than 1 or greater than 26? For which letters?
  2. Use your encoding equation to plot the \((m, c)\) pairs for all the letters in the alphabet.

    Horizontal axis, m. Scale, 0 to 26, by 2’s. Vertical axis, c. Scale -10 to 36, by 2’s.
  3. Look for the points whose \(c\) value is less than 1 or greater than 26. What letters should they be in the code? Plot the points where they should be according to the rule of your cipher.
  4. Did you end up with a graph of a piecewise function? If so, can you describe the different rules that apply to different domains of the function?

Student Response

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Activity Synthesis

Focus the discussion on the mathematical representations of ciphering and deciphering a message and their connections to functions.

Select students to share the equations they wrote for the last two questions. Display them for all to see and ask students to make some observations. Discuss with students:

  • "How are the two equations alike?" (Each pair of equations have the same letters and number.)
  • "How are they different?" (One equation seems to "undo" the other. In each pair of equations, a different variable is isolated.)
  • "Can we think of the process of encoding a message (going from \(m\) to \(c\)) as a function? Why or why not?" (Yes. For every input letter, there is only one possible output letter.) "What would be the input and output?" (In encoding a message, the plain-text letters are the inputs. The ciphered letters are the outputs.)
  • "Can we think of the process of decoding a secret code (going from \(c\) to \(m\)) as a function? Why or why not?" (Yes. Every coded letter used as an input has only one output.) "What would be the input and output?" (The ciphered letters are the inputs. The plain-text letters are the outputs.)

Explain to students that, if the rule for encoding is a function, then the rule for decoding is its inverse function. Two functions are inverses to each other if their input-output pairs are reversed, so that if one functions takes \(a\) as its input and gives \(b\) as an output, then the other function takes \(b\) as its input and gives \(a\) as an output.

Conversing: MLR2 Collect and Display. During the whole-class discussion, listen for and collect language students use to share their observations of the two equations and how they are alike and different. Write students’ words and phrases on a visual display. Be sure to emphasize words and phrases such as “undo” and “reverse.” When defining the terms inverse function, use student language from the display. Remind students to borrow language from the display and use it as a resource to draw language from during small-group and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making

15.3: U.S. Dollars and Mexican Pesos (10 minutes)


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 (\(p=19.32d\)) 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).


Give students a few minutes of quiet work time and follow with a whole-class discussion. Provide access to scientific or four-function calculators.

Reading: MLR6 Three Reads. Use this routine to support students’ reading comprehension before they begin solving problems. Use the first read to orient students to the situation. Ask students to describe what the situation is about and without using numbers (the exchange rate of U.S. dollars and Mexican pesos). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (the amount of dollars and the amount of pesos). After the third read, ask students to brainstorm possible strategies to answer the questions. The multiple reads help students connect the language in the word problem and the reasoning needed to solve the problem while keeping the cognitive demand of the task.
Design Principle: Support sense-making
Representation: Internalize Comprehension. Activate or supply background knowledge about exchange rates by inviting students to share their experiences or demonstrating an exchange.
Supports accessibility for: Memory; Conceptual processing

Student Facing

An American traveler who is heading to Mexico exchanges some U.S. dollars for Mexican pesos. At the time of his travel, 1 dollar can be exchanged for 19.32 pesos.

At the same time, a Mexican businesswoman who is in the United States is exchanging some Mexican pesos for U.S. dollars at the same exchange rate.

full-frame variety of US dollars and MX pesos
  1. Find the amount of money in pesos that the American traveler would get if he exchanged:

    1. 100 dollars
    2. 500 dollars
  2. Write an equation that gives the amount of money in pesos, \(p\), as a function of the dollar amount, \(d\), being exchanged.
  3. Find the amount that the Mexican businesswoman would get if she exchanged:

    1. 1,000 pesos
    2. 5,000 pesos
  4. Explain why it might be helpful to write the inverse of the function you wrote earlier. Then, write an equation that defines the inverse function.

Student Response

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Activity Synthesis

Discuss with students how they found the dollar value of the amounts given in pesos. Ask questions such as:

  • "How did you find the inverse function?" (By reversing the steps used to find the amount in pesos when we know the dollar amount. By solving for \(d\).)
  • "In the first function, which variable is the input and which is the output?" (Amount in dollars is the input and amount in pesos is the output.)
  • "In the inverse function, which variable is the input and which is the output?" (Amount in pesos is the input and amount in dollars is the output.)

To emphasize the two quantities switching roles, display the graphs of both functions.

Give students a moment to observe the graphs and invite them to share something they notice and something they wonder.

graph of a line with positive slope
graph of a line with positive slope

If not mentioned by students, highlight that the labels of the axes have switched places, as have the first and second values in the coordinate pair. Explain that if we trace the graphs using graphing technology, we will see that all the values of all the coordinate pairs are reversed.

Lesson Synthesis

Lesson Synthesis

To help students synthesize the key ideas in this lesson, discuss questions such as:

  • "The amount of money in cents, \(c\), is a function of the amount in dollars, \(d\). What equation can we write to represent this function?" (\(c = 100d\))
  • "How can we find the inverse function?" (We can reverse the process and solve for \(d\). To find the amount in cents, \(c\), we multiply the dollar amount, \(d\) by 100. To find the inverse, we divide the amount in cents by 100.)
  • "Why might it be helpful to find the inverse function, in this case?" (If we know an amount in cents, we can find the amount in dollars. It gives us the amount in dollars as a function of the amount in cents.)
  • "Let's say \(d=7w\) represents a function that gives the number of days, \(d\), in \(w\) weeks. A student says that the inverse function is \(w=7d\) because now the variables are switched. Do you agree? Why or why not?" (No. To find the number of days in \(w\) weeks, we multiply \(w\) by 7. So to find the number of weeks, \(w\), in \(d\) days, we need to divide \(d\) by 7, not multiply \(d\) by 7.)
  • "In general, how can you check if two functions are inverses?" (We can check if the process done to get the output of the original function gets reversed in the inverse function. If \(a\) is the input in the original function and it gives \(b\) for the output, we can see if putting \(b\) in the inverse function gives \(a\) for the output.)

15.4: Cool-down - To and From Kelvin (5 minutes)


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Student Lesson Summary

Student Facing

Sometimes it is useful to reverse a function so that the original output is now the input.

Suppose Han lives 400 meters from school and walks to school. A linear function gives Han's distance to school, \(D\), in meters, after he has walked \(w\) meters from home, and is defined by:

\(D= 400 - w\)

With this equation, if we know how far Han has walked from home, \(w\), we can easily find his remaining distance to school, \(D\). Here, \(w\) is the input and \(D\) is the output.

What if we know Han's remaining distance to school, \(D\), and want to know how far he has walked, \(w\)?

We can find out by solving for \(w\):

\(\begin {align} D &= 400 - w\\ D+w &= 400\\ w &=400 - D \end{align}\)

The equation \(w=400-D\) represents the inverse of the original function.

With this equation, we can easily find how far Han has walked from home if we know his remaining distance to school. Here, \(w\) and \(D\) have switched roles: \(w\) is now the output and \(D\) the input.

In general, if a function takes \(a\) as its input and gives \(b\) as its output, its inverse function takes \(b\) as the input and \(a\) as the output.