Yesterday I gave my students three problems. I begin today's class by having my students share their answers with the class. The first and last problems are linear functions. I will use the second problem to develop the concept of a one-to-one function. To find the inverse, I intend for my students to use a square root and to have two different answers. Often my students forget to add the plus/minus to the inverse, but I can count on one or two students to remind their classmates. If all of my students do not remember, I will ask, "What numbers, when squared, produce 4? So x^2 just equals x?"
Once we have two different answers I ask students if the inverse is a function. We discuss that a function has one answer for every domain value. Students realize that Problem 2 has two answers for every x. We then talk about notation and how f^(-1)(x) means you have an inverse function. Before concluding, I will rewrite the problem using "y=", in order to communicate to students the difference between a function and a relation.
I am now ready to discuss when an inverse is a function and what is meant by one-to-one. I start the discussion by graphing the second bell work problem. Then I ask, "How are we going to graph the inverse?" I give my students some time to think about it and come to the conclusion that we need to graph the inverse as two different functions (see page 2). Then, I graph the two functions on the same axis to give students time to think about the difference between a function and a one-to-one function.
Now, we want to think about the idea of an inverse graph as a reflection over the line y = x. To begin, I'll ask, "How do we know these 2 graphs are inverses?" Most of the time my students will realize that the original function and the other 2 functions reflect across the y=x line which was discussed yesterday. Once a student answers the question, I graph the y=x on the graph so that students can see the reflection.
I now ask students to explain why the inverse is not a function. Students explain that x-values have 2 answers. "Could we have used the original function to determine that the inverse graph would not be a function?" Students usually begin to see that looking at the y-values will help us determine if the inverse is a function. I have the students explain how to use the y-values. I write the students explanation on the board so everyone can process the response.
To bring this discussion to a good closing point, I project two graph (page 1) and two tables (page 2) and ask my students to determine if the inverse of this function is a function. I will give studnets a few minutes to work and then I will ask them to explain how they made their decision.
I hope that by this point in the class my students have arrived at the idea that the y-values of the original function can help to determine if the inverse graph is also a function. In order to help them further their thinking, I will have my students read the textbook information on the horizontal line test and the definition of a one-to-one function. Today, I will ask students to read the information individually, and, to write notes that will enable them to explain the reading later.
After students read the text, I will ask them to get into pairs. I plan to use a cooperative learning protocol:
We will complete this protocol two times.
After students discuss one-to-one. I return to the graph of Bell Work problem 2. I explain to the class that in this case we want to be able to write the inverse as a function. "What can we do so that this function is one-to-one?" Students discuss with each other for a minute. Then, I ask groups to give me ideas of what can be done.
If groups do not think about using a part of the graph, I will add x>0 to the equation and ask what this means. We then highlight the part of the graph that is defined. Some followup prompts that I may use are:
For students to understand what we have discussed, students need to practice. I plan to give my students the Are These Two Functions Inverses worksheet. The worksheet asks students to determine if two functions are inverses by using composition.
As students work I move around the room and work with students. Question 4 of the worksheet shows that 1/x is its own inverse. The last part of this question asks students to determine if every function is its own inverse and explain your reasoning. I will be looking carefully at students' responses to this question.
I will also be on the lookout for students who are struggling with their algebra techniques, such as multiplying polynomials. For students with algebra errors, I will either comment on their technique or have another student analyze the work to find the student's mistake. Having a chance to work in class gives students a chance to work together and correct each others' mistakes. I always suggest students help each other whenever possible.
As the class ends, students answer the following question on an exit slip:
Thinking about what we have learned about inverses what concept is still confusing?
This question gives me quick feedback to assess what I will need to re-address when students are reviewing for the unit assessment.