Students do run into trouble on the recursive notation in the synthesis questions. I have added a note that explains that this is a reading comprehension exercise so I'm not sure if that will change the way they approach the question or not but we'll see!

This activity took two days in class. It went pretty much as expected I think. The first day we worked on the first 3 questions and then I sent them away with 4 and 5. I talked to them a little about 5 because I thought the wording was still a little unclear. During our discussion the next day though it was nice that with the current wording this was a function and that students had a graph that they couldn't use the vertical line test on if they had just graphed a circle of radius 4 around the origin. That was unexpectedly good! Other than that they were a little disconcerted at having to fix something to get a function but they got used to that. There were some good discussions around domain and range. The second day was our work on inverse functions. During this discussion they have to be forced time and again to go back to the beginning definition and justify their reasoning from there. We believe this is very important as we come back to inverse relationships a number of times in our Calc I, Calc II sequence and each time we want them to reason from that definition. I encouraged them to think about how all their other ways of thinking about inverse functions come from that definition - switching x and y roles, arrows going from range to domain instead of from domain to range, flipping a graph over y=x, etc. They did pretty well on this overall I thought!  The graphical example is nice because they have to catch that it's not a function - this tripped up the "best" students in my class more than the students who were thinking through a lot of this stuff for the first time. It's a good point for students who have had calculus before to realize that they still need to be paying attention and being careful with what is being asked. The synthesis questions for this activity are really difficult for them as it uses recursive notation with very little intro so I think next time I would like to talk to them about those synthesis questions before they leave class and highlight that I realize this is hard thinking but that it will be very important for their learning to do that thinking without an example.