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.

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

Discuss questions such as:

• "In the inverse of function , what is the input and what is the output?" ( is now the input and is the output.)
• "The equation represents the inverse function of . 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 , we designate another variable, say , to represent it, so . How would that affect the equation for the inverse function?" It would be written as .)
• (Display the graphs of both function and its inverse.) "On the graph of , which point represents the time the tank had 80 liters of water?" (the vertical intercept, )
• "On the graph of the inverse function, which point represents that same time and same amount of water?" (the horizontal intercept, )

  

  

In this activity, students revisit another familiar situation: the percentage of homes that used only cell phones (instead of landline phones) as a function of time since 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 for ." Students are not explicitly asked to find the inverse of the 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).

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.

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 for , 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 without realizing that they have written an inverse of the original function. If so, highlight this connection and discuss how the inverse function could help answer the questions about time.