Where are the Functions Farthest Apart? - The Sequel
Lesson 7 of 11
Objective: SWBAT to use inverses to write a function that will give a horizontal distance between two functions.
Launch and Explore
Today's lesson builds upon the work we did the previous day. There is one change in the problem, finding the horizontal distance instead of the vertical distance. This very slight change produces a huge leap in terms of the rigor and challenge! Another layer is added as students must now use inverses to figure out their answer. Thus, this is a great lesson to review a concept that they learned in Algebra 2, and it allows them to see a context where the concept would be needed. Of course we do not want to tell them that they must use an inverse when they start working. They may even use them without know that they are - we will take it back to the vocabulary and notation at the end of the lesson.
Start the lesson by giving students the attached task page. I allow students to work in their table pairs and give them about 12 minutes to work on the problem. Because of our work in the previous lesson, students will probably realize that we need to use subtraction of functions, but they may get confused because of the different context.
If they are stuck, ask them how they would find the horizontal distance between the two functions when y = 0.5, for example. This will get them thinking about the need for the x-values and how to find the x-value when the y-value is known.
If a student gets the answer right away, challenge them to find a function that gives the distance between the functions at any x-value, not just the shaded region shown in the diagram. They may think that the difference of the inverse functions gives this, but sometimes the difference will be negative! This will get them thinking about the absolute value of a function.
Share and Summarize
I will start off today's class discussion just like yesterday's: I will ask a student to find the horizontal distance between the two functions on the graph when y = 0.5. Then, I will ask for the horizontal distance when y = 0.58. Then, when y = 0.3, y = 0.496, and so on and so on. Students will be even more annoyed with you than they were in the previous lesson because this is the second day of purposely being tedious. But, there is an important reason for this - it will make it very clear that we are merely subtracting the x-values! The tedium also gives us a reason to make a function for this!
Ask students about the difference between today's task and yesterday's. They will most likely point to the fact that we need to find the x-value of the functions given the y-value. Make a big deal about this and push them a little further until they get to the fact that the input and the output are switching places. Then, see if that reminds them of anything they have done in the past. Once inverses are brought into the open, you can go over the notation and procedures for finding them.
In the attached video I talk about Mathematical Practice 2 and its connection to this problem. The main difficulty of this problem is that it is very abstract. Students use inverse functions and graph the difference of them, and the x-value on the graphing calculator represents the y-value of the original graph. Make sure that the students are continuously clarifying what the symbolism means for each other; intervene when necessary to make sure that students are using language and symbols accurately.
Once we share and summarize the students' work on today's task, learners should have a good grasp of how to find the maximum horizontal distance. The problems challenge them to find the inverse function for different families of functions.
To end class, I give Problem Set as an assignment for them to work on by themselves. These tasks are from topics that they have studied before. Even so, I recommend brainstorming some things that students should do if they get stuck on a problem (look in their book, ask the teacher or a friend, Google it, etc.) Their work will give you a gauge of what they remember from Algebra 2. Exponential and logarithmic functions will be studied in a later unit, so I do not freak out if my students don't remember everything completely. My main focus is solidify their understanding of the process of producing an inverse; the algebraic steps will come later if they are not here today.
In the last problem, students will investigate the inverse of a square root function. I have asked students to graph y = x^2 and y = sqrt(x), so that they can see that the inverse function is not merely a transformation of the original function; half of it is missing! We can talk about the need to restrict the domain of the original function so that we can create an inverse function. This concept is addressed in Common Core standard HSF-BF.B.4d. A natural extension would be to think about more functions where this is the case, and then to talk about the horizontal line test.