5.1: Function Composition

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Objectives:

• Identify the inputs and outputs for a function that is the composition of two functions
• Evaluate compositions of functions at given inputs
• Find the formula for a composite function $f(g(x))$, given formulas for $f(x)$ and $g(x)$
• Find two functions $f$ and $g$ such that $h(x)=g(f(x))$, given $h(x)$

Notes:

We will NOT cover the domains of compositions of functions. Composition of functions is not easy for students. We worked with composition in 2.2, so we continue this and will also introduce decomposing functions. (As we all know, the ability to recognize functions as compositions is immensely important when it comes time to work with the chain rule in Calculus.) Students might have some familiarity with solving for actual formulas, but algebra mistakes and careless use of parentheses can cause real problems. Furthermore, they generally have not dealt with composite functions in tables and graphs, and they often struggle to have a deeper conceptual understanding of this section.

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Lesson Guide

Start class by talking about the goal of this section: We want to transform graphs by translating, reflecting, compressing, and stretching. Students may not be aware about what you mean by these transformations. A quick 2 minute demo with Desmos could help with this. (Try this premade demo.) Remark that these transformations can be understood through the perspective of compositions of functions, and so you'll spend 5.1 getting comfortable with compositions, and then you'll spend the rest of chapter 5 focusing on how to actually make these transformations happen for any given function. The demo is also included below:

Warm-Up

Option 1: Have students do Problem 1

Option 2: Have students remember what they did in \S2.2:

Let $f(x)=x+1$ and $g(x)=x^2+5$. Find
1. f(3)
2. g(4)
3. g(f(x))
4. g(f(3))

One major issue that some instructors have found with problem 1 is that it's disconnected from the rest of the section. As mentioned in the objectives, this section focuses on them finding compositions, trying to evaluate compositions, and then decomposing functions. Although we want them to be able to understand how to compose two functions given a written description of them, it's not going to be very natural for them to figure this out as the first problem of the day. You might consider saving this problem for the end, once they have a deeper understanding of the relationship between inputs and outputs with compositions. To this end, a more natural way to start this section is to just remind them that they've already seen compositions before by having them do option 2, described above. Note that if you do option 2 that you may consider skipping the next part of this lesson plan.

Identify the inputs and outputs for a function that is the composition of two functions

Initially, it may be beneficial to remind students of \S2.2. Revisit that worksheet and do some examples similar to those on Worksheet 2.2. It is probably necessary to show students the arrow diagram found in Lesson Plan 2.2 again. Then, do several examples similar to Problem 1 on Worksheet 5.1 while utilizing the arrow diagram. Your examples should emphasize giving ``practical interpretations of the functions as a complete sentence.

• Example:

• Example:

Have students do Problem 2.

Evaluate compositions of functions at given inputs

While the concept of Problem 3 is very easy, students struggle with the table having more than one function's outputs listed. Show them that the table is just a consolidation of what would be three separate tables for the functions $p(x),$ $q(x),$ and $r(x).$ Do part of Problem 3 to show them how to use the consolidated table. You may also want to do part of Problem 6 to remind them how to use graphs in a function composition.

Have students do Problem 3, 5, and 6.

Have groups of students present their answers to Problems 3 and 6.

Find the formula for a composite function $f(g(x))$, given formulas for $f(x)$ and $g(x)$

Have students do Problem 4.

Remind students to use parentheses when plugging the expression for $f(x)$ into $g(x)$ to obtain the correct formula for a function composition.

Find two functions $f$ and $g$ such that $h(x)=g(f(x))$, given $h(x)$

Now tell the students that we are going to start working backwards. Begin with a simple example of a function that can be decomposed as the composition of two nontrivial functions in more than one way. On the board, do one of the ways, and then ask students if they can find another possible decomposition.

• Example:

Have students do Problems 7 and 8. Explain to students that there are multiple correct answers to Problem 8.

Since there are multiple correct answers to Problem 8, use the remaining class time to have groups put solutions on the board.

Comments

My students found making a bubble diagram outlining the order of operations and then making a chart of possible decompositions of the function useful. For example, let $f(g(x))=h(x)=\sqrt{5x^3-3}$. The operations, in order, are cube, multiply by 5, subtract 3, and take a square root. $g$ can include any number of these operations starting from the inside and working out.

$\begin{array}{c|c|c} f(x) & g(x) & f(g(x)) \\ \sqrt{5x^3-3} & x & \sqrt{5x^3-3} \\ \sqrt{5x-3} & x^3 & \sqrt{5x^3-3} \\ \sqrt{x-3} & 5x^3 & \sqrt{5x^3-3} \\ \sqrt{x} & 5x^3-3 & \sqrt{5x^3-3} \\ x & \sqrt{5x^3-3} & \sqrt{5x^3-3} \\ \sqrt{x+2} & 5x^3-5 & \sqrt{5x^3-3} \\ \end{array} $

The last line of the table is useful for some of the WeBWorK problems and to let the students know that there are infinitely many possibilities. I would label the first and fifth decompositions with cheese so that the students know not to use any trivial decomposition in their solutions. (Morgen Bills)