# Lesson 17

Writing Inverse Functions to Solve Problems

## 17.1: Water in a Tank (5 minutes)

### Activity

In this warm-up, students make sense of a linear function with a negative rate of change, set in a familiar context. Students have seen similar relationships between the amount of water in a tank and time in an earlier unit (on solving and graphing equations). Here, they think about the quantities in terms of input and output, interpret statements in function notation, and sketch a graph of the function.

The work here prepares students to find and interpret the inverse of functions written in function notation, and to use inverse functions to solve problems.

### Launch

Arrange students in groups of 2. Provide access to scientific or four-function calculators, if requested.

### Student Facing

A tank contained some water. The function $$w$$ represents the relationship between $$t$$, time in minutes, and the amount of water in the tank in liters. The equation $$w(t) = 80 - 2.5t$$ defines this function.

1. Discuss with a partner:

1. How is the water in the tank changing? Be as specific as possible.
2. What does $$w(t)$$ represent? Is $$w(t)$$ the input or the output of this function?
2. Sketch a graph of the function. Be sure to label the axes.

### Activity Synthesis

Select students to share their responses, including their graph.

Make sure students recall that in an equation such as $$w(t)=80-2.5t$$, $$w(t)$$ is the output when the input is $$t$$, that they can interpret the equation in terms of the situation, and see why the graph is a downward-slanting line.

## 17.2: Another Look at the Tank (10 minutes)

### Activity

In this activity, students revisit the same function they saw in the warm-up, which was given in function notation. They work to find and represent the inverse function and think about what it tells us in this situation.

### Launch

Point out to students that the equations we have seen in the past few lessons use variables for the input and output. Sometimes, however, an equation that defines a function is written using function notation, so the output has the form $$w(t)$$, or $$\text {function name (input variable)}$$.

Tell students that they will now think about how to represent the inverse of a function defined using function notation.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to spark students’ curiosity about the inverse function in a situation. Display the Student Task Statement, leaving out the questions, and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving finding and interpreting the inverse function. The process of creating mathematical questions, without the pressure of producing answers or solutions, prompts students to make sense of the given information and to activate the language of mathematical questions. This work helps to prepare students to process the actual questions.
Design Principle(s): Maximize meta-awareness; Support sense-making

### Student Facing

A tank contained 80 liters of water. The function $$w$$ represents the relationship between $$t$$, time in minutes, and the amount of water in the tank in liters. The equation $$w(t) = 80 - 2.5t$$ defines this function.

1. How much water will be in the tank after 13 minutes?
2. How many minutes will it take until the tank has 5 liters of water?
3. In this situation, what information can we gain from the inverse of function $$w$$?
4. Find the inverse of function $$w$$. Be prepared to explain or show your reasoning.
5. How would the graph of the inverse function of $$w$$ compare to the graph of $$w$$? Describe or sketch your prediction.

### Activity Synthesis

Invite students to share their equation and interpretation of the inverse function of $$w$$ and their prediction of what its graph would look like.

Discuss questions such as:

• "In the inverse of function $$w$$, what is the input and what is the output?" ($$w(t)$$ is now the input and $$t$$ is the output.)
• "The equation $$t = \frac {80 - w(t)}{2.5}$$ represents the inverse function of $$w$$. What does it tell us?" (It tells us the time in minutes as a function of the amount of water in liters. It allows us to find the time that has passed if we know how much water is left in the tank.)
• "Suppose that, instead of using the function name and writing $$w(t)$$, we designate another variable, say $$r$$, to represent it, so $$r = w(t)$$. How would that affect the equation for the inverse function?" It would be written as $$t = \frac {80 - r}{2.5}$$.)
• (Display the graphs of both function $$w$$ and its inverse.) "On the graph of $$w$$, which point represents the time the tank had 80 liters of water?" (The vertical intercept, $$(0,80)$$.)
• "On the graph of the inverse function, which point represents that same time and same amount of water?" (The horizontal intercept, $$(80,0)$$.)
Engagement: Develop Effort and Persistence. Arrange the class into groups of 3–4 and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Attention; Social-emotional skill

## 17.3: Phones in Homes (20 minutes)

### Activity

In this activity, students revisit another familiar situation: a function that gives the percentage of homes with only cell phones as a function of time since year 2004. Here, the focus is on finding a linear function that models the given data and using the model to answer questions about the situation.

Previously, students saw questions about percentages, such as: “What percentage of homes had only cell phones in 2010?” Here, they see questions such as: “In what year did 50% of all homes use only cell phones?” or "Solve $$P(t)=30$$ for $$t$$." Students are not explicitly asked to find the inverse of their function, but the questions about time can be more efficiently answered by first finding the inverse of the function, motivating students to do so.

To write an equation for a possible linear model, students may find a line of best fit (and then determine its slope and vertical intercept, by hand or using technology), or they may find an average rate of change between two meaningful points on the data. Students engage in aspects of modeling (MP4) as they make such decisions.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Arrange students in groups of 2–3. Provide access to graphing technology, in case requested.

### Student Facing

In 2004, less than 5% of the homes in the U.S. relied only on a cell phone. Since then, the percentage of homes that used only cell phones have increased.

Here are the percentages of homes with only cell phones from 2004 to 2009.

years since 2004 percentages
0 4.4
1 6.7
2 9.6
3 13.6
4 17.5
5 22.7

​​​​​​

1. Suppose a linear function, $$P$$, gives us the percentage of homes with only cell phones as a function of years since 2004, $$t$$.

Fit a line on the scatter plot to represent this function and write an equation that could define the function. Use function notation.

2. Use your equation to find the value of $$P(6)$$. Then, explain what it means in this situation.
3. Use your equation to solve $$P(t) = 30$$ for $$t$$. What does the solution represent?
4. Suppose we want to know when the percentage of homes with only cell phones would reach 50%, 75%, or 100% (assuming that the trend continues and the function stays valid). What equation could be written to help us find the years that correspond to those percentages? Show your reasoning.

### Student Facing

#### Are you ready for more?

How well do you think your model will work to predict the percentage of homes with only a cell phone in future years, for example, a decade or two decades from now? Explain your reasoning.

### Anticipated Misconceptions

Students may not recall how to write an equation to model a set of data, even if they see that the points on the scatter plot appear to be linear. Ask students how they might find the rate of change in the situation, or the slope of a line that could represent the trend in the data. Prompt them with questions such as, "How quickly does the percentage of homes with only cell phones grow?" or "By how many percent, roughly, does it grow each year? How can we find out?" Once they see how to estimate a rate of change or to calculate the slope of a line that fits the data, ask what other information they might need to write a linear equation.

### Activity Synthesis

Invite students to share how they wrote an equation to model the relationship in the data. Then, focus the discussion on how they answered questions about time, such as how they solved $$P(t)=30$$ for $$t$$, or found the years in which the percentages of cell-phone-only homes reached 50%, 75%, and so on.

When writing an equation for the last question, students may have solved for $$t$$ but not realize that what they have done is writing an inverse of their original function. If so, highlight this connection and discuss how the inverse function could help us answer the questions about time.

Speaking, Listening: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each strategy that is shared about how students answered the questions about time or found the years when the cell-phone-only homes reached different percentages, press for details in students’ explanations by requesting for students to challenge an idea or elaborate on an idea. Encourage students to connect their strategies multi-modally by using different types of sensory inputs, such as using gestures or talking about the context of what is happening. Throughout this discussion, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Optimize output (for explanation); Support sense-making

## Lesson Synthesis

### Lesson Synthesis

Discuss with students:

• "In this lesson, we saw some real-world situations where it was useful to find the inverse of a function. What kinds of problems does the inverse of a function allow us to solve? How do they help us understand a situation?"

## Student Lesson Summary

### Student Facing

The water in a rain barrel is being drained and used to water a garden. Function $$v$$ gives the volume of water remaining in the barrel, in gallons, $$t$$ minutes after it started being drained. This equation represents the function:

$$v(t) = 60 - 2.25t$$

From the equation and description, we can reason that there were 60 gallons of water in the rain barrel, and that it was being drained at a constant rate of 2.25 gallons per minute.

This equation is handy for finding out the amount of water left in the barrel after some number of minutes. In other words, it helps us find the output, $$v(t)$$, when we know the input, $$t$$.

Suppose we want to know how long it would take before the barrel has 20 gallons of water remaining, or how long it would take to empty the barrel. Let's find the inverse of function $$v$$ so that the volume of water is the input and time is the output.

Even though the equation is in function notation, we can still solve for $$t$$ as we had done before:

\begin {align} v(t) &= 60-2.25t\\v(t) + 2.25t &= 60\\ 2.25t &= 60-v(t)\\ t &=\dfrac{60-v(t)}{2.25} \end{align}

This equation now shows $$t$$ as the output and $$v(t)$$ as the input. We can easily find or estimate the time when the barrel will have 20 gallons remaining or when it will be empty by substituting 20 or 0 for $$v(t)$$, and then evaluating $$\frac{60-20}{2.25}$$ or $$\frac{60-0}{2.25}$$, respectively.