Lesson 16

Earlier in the course, students have spent some time writing equations in one variable, two variables, and multiple variables to represent relationships and constraints. They have also rearranged equations to isolate a particular variable. This warm-up activates that prior knowledge, preparing students to write equations for linear functions that involve two or more operations, and then to write their inverses.

To ensure access, the equations students are expected to write are scaffolded—starting with one operation and moving to two operations, from using numbers and letters to using only letters. The scaffolding also allows students to see structure in the equations, which would in turn be helpful when they try to reverse the process.

Invite students to share their equations. Record and display them for all to see. Ask students who wrote the second set of equations how they went about doing so. Highlight two main points:

• Writing the second set of equations essentially involves undoing each operation in the first equation and doing so in reversed order.
• Each equation in the second set is the inverse of the corresponding equation in the first question. Each isolated variable is now an output, while previously it was an input. The variable that was previously isolated is now an input.

In this activity, students investigate inverse functions in the context of temperature scales.

Students are given an equation that defines the temperature in degrees Fahrenheit as a function of the temperature in degrees Celsius (\(F = \frac59 C+32\)) and are asked to find some temperatures in both units. Using this equation to find a temperature in Celsius involves substituting the temperature in Fahrenheit for \(F\) and solving the equation. Students do so several times over. The regularity in repeated reasoning with numerical values (MP8) paves the way for students to write the equation for the inverse function, which entails solving for \(C\) before substituting values of \(F\).

Students are also prompted to verify that two unfamiliar equations represent inverse functions. To do so, students need to analyze the structure of one equation, use it to reverse the process that defines the function, and see if the reversal leads to the other equation (MP7).

Explain to students that the Rankine scale is a temperature scale that is sometimes used in engineering systems, typically alongside measurements in Fahrenheit.

Provide access to calculators, in case requested.

Some students may struggle to solve for \(C\) because of the rational coefficient of \(C\) in \(F=\frac95 C + 32\). They may be able to rearrange the equation to \(F-32=\frac95 C\) but then get stuck at that point. Some strategies for supporting their reasoning:

• Think of \(\frac95\) simply as a number being multiplied to \(C\), so to find \(C\), we can divide both sides of the equation by \(\frac 95\). (Students can still reason this way even if they do not recall or know that dividing by \(\frac95\) gives the same result as multiplying by \(\frac59\). Their resulting equation may look like: \((F-32) \div \frac95 = C\).)
• Think of \(\frac95 C\) as \(9 \boldcdot \frac15 \boldcdot C\) and then undo one operation at a time.
• Write \(\frac95\) as a decimal, 1.8, and divide both sides of the equation by 1.8. (Clarify that, depending on the fraction, rewriting as a decimal may not always be the most helpful way to go.)

In the course of solving for a variable, students may neglect to apply the distributive property correctly when multiplying a number to an expression. For example, when performing the last step to solve \(F-32=\frac95 C\) for \(C\), they may write \(\frac59F - 32=C\), instead of \(\frac59 (F-32)\). Remind students that any operation performed on both sides of an equation needs to be applied to all the parts on each side.

Discuss with students:

• "How did you find the inverse of the first function?" (By solving the equation for \(F\). By noting what is done to the temperature in Celsius to get the temperature in Fahrenheit and then reversing the steps.)
• "Suppose we want to find the temperature in Celsius when it is 10 degrees Fahrenheit and when it is 90 degrees Fahrenheit. Which equation would you use? Why?" (The second equation, because it doesn't involve solving an equation, which tends to be quicker.)
• "How did you verify that the two equations that relate the temperature in Rankine and the temperature in Celsius are inverse functions?" (By solving the first equation for \(C\) and see if it matches the second equation, or by solving the second equation for \(R\) and see if it matches the first equation.)
• "What does each inverse function tell us?" (The first one tells us how to find the temperature in degrees Celsius if we know the temperature in degrees Fahrenheit. The second tells us how to find the temperature in degrees Celsius if we know the temperature in degrees Rankine.)

If time permits, display the graphs of the two functions relating temperature in Celsius and in Fahrenheit. Emphasize how the input and output variables are switched.

The two points on each graph mark the boiling temperature and freezing temperature. In both graphs, the boiling temperature, \(100 ^\circ C\), is paired with \(212 ^\circ F\), and the freezing temperature, \(0 ^\circ C\), is paired with \(32^\circ F\), but the \(C\)- and \(F\)-values show up in a different order in the coordinate pairs.)

This info gap activity gives students an opportunity to determine and request the information needed to evaluate and find the inverse of a function.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Display both equations for all to see and ask students to compare them. Discuss questions such as:

• "Do both equations define functions? How can we tell?" (Yes. For each price per mug, there is only one possible number of mugs that can be bought. For each number of mugs that the club buys, there is only one possible original price.)
• "In which situation is the first equation more useful?" (When we know the price and want to know how many mugs we can buy.)
• "In which situation is the second equation more useful?" (When we know the number of mugs bought and want to know what the original price was.)
• "If so, are they inverse functions? How do you know?" (Yes, the two equations “undo” each other. The input of the first function is the output in the second function. The output of the first function becomes the input in the second function.)

Highlight for students how solving the first equation for \(p\) gives us the second equation, and solving the second equation for \(n\) gives us the first equation.

This optional activity allows students to practice writing and interpreting inverse functions. It also prompts students to consider the domain and range of a function and of its inverse, and to do so in terms of a situation. Students think about the set of possible outputs of an original function and then consider, based on what the quantities represent, whether the same set of values makes sense as the domain of the inverse function.

Invite students to share their response and reasoning. Make sure students see that because the number of seats, \(S\), is found by multiplying the number of table, \(n\), by 4 and then adding 2, the inverse function can be found by subtracting 2 from \(S\) and then dividing the result by 4.

Discuss with students:

• "Why are 23 tables needed for 94 guests but 24 tables needed for just 1 more guest?" (Entering 95 for \(S\) in the equation \(n=\frac{S-2}{4}\) and gives \(n=23\frac14\), but a fractional value of \(n\) does not make sense in this situation, so one more table is needed, though there will be open seats.)
• "The range of the first function, the set of \(S\)-values, are numbers that are 2 more than multiples of 4 (6, 10, 14, . . .). In the inverse function, is the domain limited to the same set of \(S\)-values? Or are numbers like 17 and 85 also possible inputs?"

Highlight that, just as solutions to equations need to be interpreted in context, so do the domain and range of a function and its inverse.

• The equation for the first function, \(S=4n+2\), tells us the number of seats at \(n\) tables, assuming all the tables are filled completely and regardless of whether the seats are occupied by guests. The range values of 6, 10, 14, . . . reflect these assumptions.
• The equation for the inverse function, \(n=\frac{S-2}{4}\), gives the number of tables if we know the number of seats, \(S\), but we don't assume that the seats completely fill each table. The domain of the inverse function (the set of \(S\)-values) could include any whole number (not just 6, 10, 14, . . . ). When an input gives a fractional number of table for the output (such as \(n=23\frac14\)), we need to interpret it in terms of the situation.