This lesson is scaffolding for proving two functions are inverses. My students briefly worked with composition of functions last year, but the transition to our Common Core Curriculum caused this concept to be covered too quickly. As we continue to implement the curriculum this will be a quicker review included in the lesson on inverses.
In Algebra 2 students did work with problems similar to today's Bell Work. Many students will remember a similar problem. I have changed the problem a little so that students have to think about what is happening. Usually the question is what is better for the customer. I have changed the context so that students are asked to determine what is better for the store. I also require the students to verify their answer which will keep the students from repeating what was done last year.
To begin the lesson, I pose this question to the students:
Our class has been asked to help Kohl's to determine which process would be the best for the store. A customer can turn in two discount cards per purchase. The two discounts are a 30% off the total and the other discount is a $10 Kohl's cash certificate.
Would it be better for the store to take the 30% off then deduct $10 or deduct $10 then reduce the total by 30%?
Show your work that verifies your decision.
I will let students work on the Bell Work task for about 5 minutes. As they do I walk around the room looking at work and asking questions like, "Are you sure?" If a student is stuck I will say something like try to find out which is better if a $100 item is bought. After five minutes have passed I bring the class back together to discuss the problem.
As I walked around the room, I have identified students that did the problem using an example. I will have one of those students share their work on the board. If I have students who used different values I have both share the work. If I don't have 2 different amounts, I ask would the result be the same if we used a different amount? We will work the problem using a different amount to verify. It often shocks the students to see that the difference is the same for the 2 amounts.
Instead of just giving students the notation and definition for composition. I develop the notation through the bell work problem. This is explained in this video.
After developing composition of functions with a real world example, students are ready to learn the more formal version of composition. I project the definition of composition on the board. I give students a few minutes to read the information on the board. Students may have seen f(g(x)), but the other notation is new. We look at Figure 1.9 and discuss how you find the f(g(x)) means you find the answer (output) for g then put that answer in as the domain (input) for the function f.
Now, I give students g(f(x)) and we write it in the new notation. I ask if g(f(x)) is different from f(g(x))? Will the results be the same? Students explore and discuss the results.
In order to give students the opportunity to practice with the notation, we will do some work using tables next. The students work on the given problems and we discuss each answer. At first, students are confused about where to start. I ask how you start with order of operations. Once they remember to look at the function on the inside, I will ask, "What does g(-2) mean? How can we evaluate g(-2)?" Despite their prior work with functions, I expect some students will need more guidance. Oftentimes, once a student finds g(-2), they are not sure what to do with that answer. Today, I will remind students how we did the Bell Work.
After working with the tables, we will move on to graphs. Doing composition with graphs allows me to review how the output is the y-value or in this case the value of f(x) or g(x) for a given value of x. Even though my students have graphed functions for several years, some still have trouble remembering that the output is plotted on the vertical axis.
By working with tables and graphs, students can better see the structure of a composition of two functions. In particular, they can observe how the range of the inside function becomes the domain of the outside function. When the class goes over the answers for the graphs, I use my fingers to show how to read the information.
To extend this activity, I will ask my students to graph f(x), g(x), f(g(x)) and g(f(x)) on the same coordinate plane. I will ask them to observe and discuss how composition creates a new function by describing the function in terms of its plot.
To end class today students are given an the following question to answer:
Is there a difference between f o g (x) and f(g(x))? Explain.
Many students forget how important parenthesis are in mathematics. This question lets me see which students realize the parenthesis gives you a different function operation. I will also know the students who will struggle when we work with transformed trigonometric functions.