Finding the Inverse of a Function Day 2 of 2
Lesson 11 of 15
Objective: SWBAT find and verify that functions are inverses.
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.
Is the inverse a function?
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.
One to One Functions
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:
- Student 1 talks for two minutes explaining the information in the text to their partner. Student 2 listens and takes notes about what is said, but may not respond or question during the initial two minutes.
- After 2 minutes Student 2 gets one minute to rephrase what was said, ask questions or make corrections. Student 1 is now the listener.
- After both students have talked a final 30 seconds is provided for students to clarify any lingering questions.
We will complete this protocol two times.
- We begin with the horizontal line test. Students are told to focus on the use of the horizontal line test and how to use the test to make decisions. Once the class finishes the cooperative activity I bring the class together to debrief. I ask "What is the use of the horizontal line test?" and "How does it work?"
- We use the same cooperative activity for the definition of a on-to-one function. Before beginning this part of the cooperative activity students are given the opportunity to review the information from the text. Many students want to make better notes after the first discussion. The pairs are to focus on what it means for a function to be one-to-one? If a function is one-to-one what can be determined about its inverse?
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:
- Is this function now one-to-one?
- Will we use the entire inverse? What part of the inverse will we use? Is the inverse now a functions?
- If I change the domain to x<0 how will that change the inverse?
Practice verifying inverses
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.