I start today with a problem reviewing yesterday's lesson. The graph has f and g graphed on the same coordinate plane. The graph is slightly different from yesterday's work. Students may be confused at first, but I will ask students to talk among their group to determine how to deal with the situation. After students work independently for a few minutes, I will have them share their plans and results.
As we go over the student responses, I will make comments to with respect to finding the output for for g and using that as the input, x, for f to determine the final answer. After reviewing student responses, I will ask the students to write out an explanation for how to find f(g(x)). Afterward, I will share a statement that provides a complete explanation.
After revising this statement, I ask my students to revise the statement to put the meaning into words that they think will be easier to understand. We then draw a diagram of what is done and put information about the domain and range under the diagram. I then have students explain how to find g(f(x)). After explaining the process students draw a diagram for finding g(f(x)).
Developing a deeper understanding of the process of how a composition is evaluated helps students understand how to use composition to interpret the structure of a function, if it can be treated as a composition of simpler functions. Students who move on to Calculus need to be comfortable thinking in this way in order to understand the Chain Rule for derivatives and the Substitution Method for Integration.
Today, the class is moving from using tables and graphs to evaluate a composition to finding the composition when f and g are functions.
To begin I give students 2 problems to evaluate. As the students are working I move around the class observing student work and listening to group discussions. Some groups do not know where to begin. I will ask questions to guide them to work the problem. My questions include:
I also have struggling students get the diagram out that was made in the bell work. We look at that to help guide them with my questions.
Once the students have found the answer. I give students the same functions, but ask to find f(g(x)). Students work in groups for a few minutes. I bring the class back together. I ask if any group has a possible answer. Those answers are shared. I then question how the students determined the answer.
As we discuss the process I draw a diagram (pg. 2) similar to the bell work diagram. The students are explaining as I draw. We discuss how the range for g is the value we get by evaluating x-1 so I put x-1 in the middle circle. This is now the domain used in f. We then finish the diagram. As we draw the diagram we also determine the range of the f(g(x)).
After finding f(g(x)) the students find g(f(x)). I have students find both compositions to show that the commutative property does not work for composition.
To close this segment of the lesson, I give students another problem. As students work I check students work. Again we share the answers with the class.
For calculus it is very important for students to be able to determine what functions have be composed. This idea is used when students use the Chain Rule for derivatives and the Substitution Method for Integration.
I begin working with decomposing a composition by letting the students think about a problem. At first students are not sure what do do. The issue is understanding the question. I help students dissect the question with questions such as:
Once we have looked at what the students are to find I give students a minute or two to try and find the answer. Students share possible answers. As we put answers on the board I have the class check to see if the expressions for f(x) and g(x) will produce h(x).
I begin with an easier composition, so that the students can quickly see the two functions. I give students some more problems to practice. The last two problem (pages 2-3) have multiple answers. Student may think an answer is wrong once one correct answer is found. I ask for other answers. If know one suggests another answer I will say does this work and put up an answer that works but has not been shared. This will usually bring a response like, "That is what I had." Students need to know that some activities we do have multiple answers. As the class progress students are more willing to share when their method or answer is different than others.
Once students have seen different ways to work with composition I share another contextual problem, this time a problem from physics. I scaffold the problem by asking students to write functions for the radius and the area. This problem is similar to yesterday's Bell Work.
I ask my students to work on this problem in groups. I move around the room as students work and after a few minutes I start choosing students to share their answers on the board. After a student shares an answer, I will pick a different student to explain how the answer can be found.
I assign page 88, #42, 48, 52, 64, 72 from "Precalculus with Limits" by Larson for homework. The problems have the students practice finding compositions, decomposing composition and interpreting the meaning of a composition in a problem.
To end class, students complete a problem before leaving class. This problem has 2 purposes. First, I am able to assess student understanding of finding a composition. The second purpose is to have students think about when f(g(x))=g(f(x))=x. Some students will realize that the 2 functions are inverses. They may or may not remember the word inverse but they will know one function undoes the other function.
The next topic we discuss is inverses and we use composition to verify 2 functions are inverses.