<p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">Students will learn about the concept of composition of functions in this lesson. Students will:&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- evaluate a composition&nbsp;of functions based on an input.&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- use a set of graphs to evaluate a composition of functions.&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- identify the function that is made when two functions are put together.</span></p>

<p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- How may functions be built upon one another?</span></p>

<p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- Input: In functions, the independent variables.&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- Output: In functions, the dependent variable.</span></p>

<p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- Function Machine cards (M-A2-6-3_Function Machine)&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- It Takes Two Worksheet and KEY (M-A2-6-3_It Takes Two and M-A2-6-3_It Takes Two KEY)&nbsp;</span></p>

<p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- The Function Machine activity measures students' comprehension of the composition process. To obtain the correct output, each operation must be performed in the precise order. The proper process begins with examining the arithmetic of each computation. Some students will execute the function operation erroneously, using the last output rather than the original function.&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- The It Takes Two worksheet will be assessed to determine whether students are using the proper order of composition. Before students retry the composition, check the order of composition for each wrong result and remind them of the inside-out rule.</span></p>

<p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">Active Engagement, Modeling, and Explicit Instruction&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">W: This lesson teaches students how to visually express function compositions and evaluate their outcomes.&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">H: The function machine cards are a familiar notion for many students from primary school. The activity reminds students of the well-known input/output process, as well as the necessity that there be exactly one output for each unique input in order for the relation to be a function.&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">E: By focusing on the language of composition, students can develop their own and collective understanding of what it means to compose. Writing prose, composing music, and drawing a picture are all examples of compositions that necessitate the incorporation of certain individual pieces as component portions of the final product. Emphasizing the precision of speaking and writing the composition is also critical for promoting a more full understanding.&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">R: After completing the It Takes Two assignment, students should reflect on three key questions about function composition. What exactly does a function do? What exactly does composition mean? Does the sequence of composition matter?&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">E: Evaluate each student's responses to the three questions and the language employed. Does the student's response indicate that he or she has received a personal understanding adequate to create an original representation? Does the explanation of why the order of function composition is important include an example that displays adequate understanding? Is the student able to provide further questions concerning the properties of composing functions that can help with a more broad or specialized understanding?&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">T: This lesson emphasizes kinesthetic learning. The kinesthetic and social learners in the room will enjoy and remember their time at the front of the room, on stage, and performing arithmetic with their classmates. Other students in the class can also contribute by providing verbal input. Visual learners will benefit by color-coding the graphs of the two functions during the It Takes Two task. Ensure that colors are available for individuals who require them. While students are working in groups on the worksheet, identify individuals who require additional assistance comprehending the concept of composition and assist those students in verbalizing their questions to their group, facilitating small-group conversations,&nbsp;and ensuring comprehension.&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">O: The lesson starts with a basic idea that students are already familiar with, but is necessary for today's subject (functions). They then begin to work on creating simple function compositions; all of the functions offered in the Function Machine are relatively simple operations, allowing students to focus on the composition portion of the activity rather than becoming bogged down in complex calculations. The lecture encourages students to investigate the new topic on their own, both by supplying inputs and by attempting to construct algebraic rules for composition. The lesson also allows students to work independently on the It Takes Two worksheet, which covers both many representations of functions and the concept of composition.&nbsp;</span></p>

<p><strong>"Every day, we apply the composition of functions to solve mathematical problems by using the solution to one problem as the input for another. For example, suppose you have $20 and want to go watch a movie for $6.50, then eat dinner for $9, and finally buy a gift for your brother. Of course, a single linear function can represent all of these, but in more complicated situations, composition may be more practical (and can produce the linear function that represents all of these situations)."</strong></p><p>Ask for volunteers. Begin with one person on the board. Allow the student to secretly select one of the functions from the laminated Function Machine cards. (M-A2-6-3_Function Machine)</p><p>Remind students that functions take an input and return an output. Have students volunteer inputs to the student with the chosen function, and then have that student apply the function to the input and display the results on the board. Maintain a table of x- and y-values (inputs and outputs) for the chosen function on the board. Have students keep guessing until they can guess the function.</p><p><strong>"That's an excellent overview of how 'regular' functions work: you provide an input and receive an output. Today, we'll discuss composing functions. What does the word compose mean, and where have you heard it before?"</strong> (composing an e-mail, writing a composition, being a music composer, and so on).</p><p><strong>"All of those uses of the word compose have essentially the same meaning—to put together. That is exactly what we are doing when we compose functions; we will put them together, although in a very specialized way."</strong></p><p>Have two more volunteers come to the front of the class and secretly choose a function from the Function Machine cards. Name each function (f, g, etc. You can also use each student's first initial.) Explain to the class how we will combine, or compose, the two functions.</p><p><strong>"First, we will provide the first function with some input. The output of that function will then be used as the input for the next function."</strong></p><p>Demonstrate this to students first using the Function Machine (M-A2-6-3_Function Machine), showing them the functions <i>f(x)</i> and <i>g(x)</i> on the front of the card, then flipping it over to show them <i>f(g(x))</i> and <i>g(f(x))</i>. Emphasize that whatever the function <i>g(x)</i> is, you place the full function in place of the <i>x</i> in <i>f(x)</i> for<i> f(g(x))</i>, as that notation actually means to substitute <i>g(x) </i>for <i>x</i> in <i>f(x)</i>. Also, the order is important; <i>f(g(x))</i> will not be the same as <i>g(f(x)) </i>in most circumstances. Show the two index cards and ask students to notice how different the findings are. Once you think students grasp this, continue with the following examples:</p><p>First, try some instances (without worrying about recording the outcomes). Have students give the first function an input, then have the person representing that function tell the other function/student the output, and then have the second person provide the "final output."</p><p>Get two new students with two new roles. Again, give them function names and explain the notation for function composition. Stress which of the two functions the class will send the initial input to (for example, f) and the order in which the functions are expressed under composition, i.e., <i>g(f (x))</i>, with the first function on the inside of the composition statement. Also, use the alternate notation for composition, <i>(g ○ f)(x)</i>. Emphasize that how created functions are written and spoken is critical to producing accurate representations of the functions being composed. Ask students say and speak the terms appropriately. Write <i>g(f(x))</i> and have them practice saying it correctly: "<i>g</i> of <i>f</i> of <i>x</i>."</p><p>Create an input-output table for <i>g(f(x))</i>, allowing the class to supply inputs while the functions provide the necessary output. Now, ask the students how calculating <i>f(g(x))</i> differs from calculating <i>g(f(x))</i>. Remind students of the significance of order while writing the functions.</p><p>Create an input-output table for <i>f(g(x))</i> and compare it to the original one.</p><p><strong>"Does the order in which we combine functions produce any significant differences?"</strong> (Yes, most of the time.)</p><p>Depending on how the class is doing (and how much time remains), this activity can be repeated to allow for composition exploration, such as using more than two students/functions, attempting to find functions where composition makes no difference, allowing students to create their own functions instead of using the Function Machine cards, and so on.</p><p>Finally, return the scenario to having two functions, both selected from the Function Machine cards. Remind students that at the start of class, they submitted inputs to a function, received outputs, and ultimately determined what the function was. Repeat the process with both functions, <i>f</i>, and <i>g</i>, and write <i>f</i> and <i>g</i> on the board along with their rules in function notation.</p><p><i>f (x) = x + 2 and g(x) = 3x</i></p><p>Students should contribute inputs to the composition (image) and work together to create outputs in the correct order. Record the inputs and outputs, and ask them to come up with a single rule that applies to the entire composition rather than two separate rules. Guide students through a few examples (including reversing the order of composition) of creating a single rule to cover the composition before moving on to the abstract, algebraic approach<br>&nbsp;</p><p>Example:<i> f (x) = x + 2; g(x) = 3x</i></p><p><i>(g ○ f )(x) = 3(x + 2)</i></p><p><i>f(x) = x + 2; g(x) = 3x</i></p><p><i>(f ○ g)(x) = 3x + 2</i></p><p>When using the algebraic approach, remind students that whatever is inside in parentheses serves as our input, and we replace all instances of <i>x </i>with it before simplifying.</p><p><strong>"Now that we've investigated composition with functions represented as equations and with the Function Machine, let's look at composition with graphs. Remember, however, that graphs are essentially representations of functions (equations) and tables, therefore, the concept will be almost the same."</strong></p><p>Distribute the It Takes Two exercise (M-A2-6-3_It Takes Two and KEY) and work with students on the first few problems. Again, begin with the inside function, obtaining the <i>y</i>-coordinate associated with the supplied <i>x</i>-coordinate, and then use that <i>y</i>-coordinate as the <i>x</i>-coordinate for the second, outer function. Work with the class through as many instances as needed; once individual students understand, they will be able to work ahead at their own pace.</p><p>Use the questions below to help students reflect on the lesson:</p><p><strong>"What does a function do on its own?"</strong> (<i>take an input, provide an output, modify a number</i>)</p><p><strong>"What does composition of functions mean?"</strong> (<i>to merge two functions, apply one's output as the other's input</i>)</p><p><strong>"Is the order of composition significant?"</strong> (<i>yes</i>)</p><p><strong>Extension:</strong></p><p>Assign students to compose two extra compositions to present to the class. Each student will work through each composition, demonstrating each step and the corresponding arithmetic operation. Students should be prepared to explain what and why for each stage, as well as demonstrate the method individually to the full class.</p><p>After several students have displayed a solution to each of their problems, ask the class to identify the aspects of the questions that make certain solutions more difficult than others.</p><p>Some possible answers may include:</p><p>functions which use non-integers<br>functions involving exponents and/or radicals<br>multiple compositions</p>