# Engaging students: Computing the composition of two functions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Alexandria Johnson. Her topic, from Algebra II/Precalculus: computing the composition of two functions.

The following link is to a worksheet over composition of functions. The worksheet allows students to explore composition of functions without outright telling them what composition of functions is. Instead, the students are working on real world problems about shopping in a store that is having a 20% sale with mystery coupons. In the worksheet, students explore whether or not it matters which discount is applied first and the equations that go along with each scenario. This worksheet is interesting because it approaches composition of functions in an explorative way and it is using a real-world situation students in high school may find relatable, which can help hook students that are math-phobic.

https://betterlesson.com/community/document/1326462/going-shopping-student-materials-docx

Computing the composition of two functions requires prior knowledge of basic operations and combining like terms. This topic will expand upon their knowledge of basic operations by applying them to functions. Students will be able to add, subtract, multiply, and divide functions. Students should be able to use the distribution property; this is important when students are writing (fog)(x) and (gof)(x). During this topic, students should be able to expand upon their knowledge of creating functions from real world problems, which can be seen in the worksheet from the link above.

Musical composition is a way this topic can appear in high culture. Musical composition is the process of combining notes, chords, and melodies in a particular way. Arranging the notes, chords, or melodies in different ways can change the composition. Function composition is the combining of different functions f(x) and g(x) in different ways like addition, subtraction, multiplication, and division. Order usually matters in function composition just like in musical composition. If you have several band students, or musically inclined students, this would be a good hook to grab students interest.

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