Activity Guide
Purpose: To define functions and inverse functions and learn how to work with these definitions. (Also domain, range and functional notation are introduced.)
Classroom Procedure: We begin this activity by defining functions during a class discussion. I usually use something like "A function is a way of describing a relationship between two variables - one is the input variable and the other is the output variable - and the function must associate each value of the input variable with exactly one value of the output variable". After this definition is established students are asked to try each of the problems that follow on their own first and then compare their answers with classmates nearby. While they are working on these problems the instructor is walking around the room monitoring progress, encouraging students to try the problems and answering any questions they might have (without telling them how to do the problem). After they have done this we discuss the problems as a class. This class discussion might happen after each problem or after they have worked on a couple of problems if thinks seem to be going fairly well.
Ideas this Activity Builds On: This is the first activity we have that follows up from the introductory bottle graph activities. In those activities we have dealt with functions informally, we have mentioned linear relationships because this is something that most students are familiar with. We have talked about slope and vertical intercept for the linear relationships and general descriptions of growth and decay (including concavity) for the non-linear relationships. Now we are introducing the formal definition of functions and inverses functions.
Introduction/Motivation of the Activity: During the first two activities we have actually been working on functions that we never explicitly talked about. After we have discussed the definition of a function as a class, we can connect back to the bottle graphs to ask if they were functions and why. Then continue on to the rest of the activity.
Need to Establish by the End of Activity/Wrap-Up: By the end of this activity students need to understand the definition of function well enough to identify whether any given relationship (in any form) is a function or not. They need to be able to use the definition of inverse function to determine the inverse of a function given in any form.
Additional Notes: The synthesis questions for this activity push students to understanding how recursive notation can be used to describe the manner in which a function is growing or decaying. Students find this very difficult but it is a great activity in reading a mathematical statement and making practical sense of it. We find that our students struggle with reading throughout this class and this is a good first place to start talking to them about how important it is that they are proficient readers of mathematics by the end of the semester. (As many application problems require students to be proficient readers to even start the problems.)