# Composition of Functions (Lesson 4.5)

###### Unit 1: Sequences and Linear Functions

**Day 1:**Recursive Sequences**Day 2:**Applications of Arithmetic Sequences**Day 3:**Sum of an Arithmetic Sequence**Day 4:**Applications of Geometric Sequences**Day 5:**Sequences Review**Day 6:**Quiz 1.1 to 1.4**Day 7:**Linear Relationships**Day 8:**Point-Slope Form of a Line**Day 9:**Standard Form of a Linear Equation**Day 10:**Quiz 1.5 to 1.7**Day 11:**Unit 1 Review**Day 12:**Unit 1 Test

###### Unit 2: Linear Systems

**Day 1:**Linear Systems**Day 2:**Number of Solutions**Day 3:**Elimination**Day 4:**Larger Systems of Equations**Day 5:**Quiz 2.1 to 2.4**Day 6:**Systems of Inequalities**Day 7:**Optimization Using Systems of Inequalities**Day 8:**Quiz 2.5 to 2.6**Day 9:**Unit 2 Review**Day 10:**Unit 2 Test

###### Unit 3: Function Families and Transformations

**Day 1:**Interpreting Graphs**Day 2:**What is a function?**Day 3:**Translating Functions**Day 4:**Quiz 3.1 to 3.3**Day 5:**Quadratic Functions and Translations**Day 6:**Square Root Functions and Reflections**Day 7:**Absolute Value Functions and Dilations**Day 8:**Equations of Circles**Day 9:**Quiz 3.4 to 3.7**Day 10:**Unit 3 Review**Day 11:**Unit 3 Test

###### Unit 4: Working with Functions

**Day 1:**Using Multiple Strategies to Solve Equations**Day 2:**Solving Equations**Day 3:**Solving Nonlinear Systems**Day 4:**Quiz 4.1 to 4.3**Day 5:**Combining Functions**Day 6:**Composition of Functions**Day 7:**Inverse Relationships**Day 8:**Graphs of Inverses**Day 9:**Quiz 4.4 to 4.7**Day 10:**Unit 4 Review**Day 11:**Unit 4 Test

###### Unit 5: Exponential Functions and Logarithms

**Day 1:**Writing Exponential Functions**Day 2:**Graphs of Exponential Functions**Day 3:**Applications of Exponential Functions**Day 4:**Quiz 5.1 to 5.3**Day 5:**Building Exponential Models**Day 6:**Logarithms**Day 7:**Graphs of Logarithmic Functions**Day 8:**Quiz 5.4 to 5.6**Day 9:**Unit 5 Review**Day 10:**Unit 5 Test

###### Unit 6: Quadratics

**Day 1:**Forms of Quadratic Equations**Day 2:**Writing Equations for Quadratic Functions**Day 3:**Factoring Quadratics**Day 4:**Factoring Quadratics. Part 2.**Day 5:**Solving Using the Zero Product Property**Day 6:**Quiz 6.1 to 6.4**Day 7:**Completing the Square**Day 8:**Completing the Square for Circles**Day 9:**Quadratic Formula**Day 10:**Complex Numbers**Day 11:**The Discriminant and Types of Solutions**Day 12:**Quiz 6.5 to 6.9**Day 13:**Unit 6 Review**Day 14:**Unit 6 Test

###### Unit 7: Higher Degree Functions

**Day 1:**What is a Polynomial?**Day 2:**Forms of Polynomial Equations**Day 3:**Polynomial Function Behavior**Day 4:**Repeating Zeros**Day 5:**Quiz 7.1 to 7.4**Day 6:**Multiplying and Dividing Polynomials**Day 7:**Factoring Polynomials**Day 8:**Solving Polynomials**Day 9:**Quiz 7.5 to 7.7**Day 10:**Unit 7 Review**Day 11:**Unit 7 Test

###### Unit 8: Rational Functions

**Day 1:**Intro to Rational Functions**Day 2:**Graphs of Rational Functions**Day 3:**Key Features of Graphs of Rational Functions**Day 4:**Quiz 8.1 to 8.3**Day 5:**Adding and Subtracting Rational Functions**Day 6:**Multiplying and Dividing Rational Functions**Day 7:**Solving Rational Functions**Day 8:**Quiz 8.4 to 8.6**Day 9:**Unit 8 Review**Day 10:**Unit 8 Test

###### Unit 9: Trigonometry

**Day 1:**Right Triangle Trigonometry**Day 2:**Solving for Missing Sides Using Trig Ratios**Day 3:**Inverse Trig Functions for Missing Angles**Day 4:**Quiz 9.1 to 9.3**Day 5:**Special Right Triangles**Day 6:**Angles on the Coordinate Plane**Day 7:**The Unit Circle**Day 8:**Quiz 9.4 to 9.6**Day 9:**Radians**Day 10:**Radians and the Unit Circle**Day 11:**Arc Length and Area of a Sector**Day 12:**Quiz 9.7 to 9.9**Day 13:**Unit 9 Review**Day 14:**Unit 9 Test

#### Learning Targets

Given two functions, compose new functions by inputting one into the other.

Evaluate a composition of functions for given input values.

Tasks/Activity | Time |
---|---|

Activity | 20 minutes |

Debrief Activity with Margin Notes | 15 minutes |

QuickNotes | 5 minutes |

Check Your Understanding | 10 minutes |

#### Activity: The Pumpkin Pi Bakery - Part 2.

#### Lesson Handouts

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#### Answer Key

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#### Homework

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#### Experience First

We're revisiting the Pumpkin Pi bakery today. Since students are familiar with this context we are hoping they will be able to focus on the composition and less on writing the original functions. We will be using the same function for profit based on the number of pies produced as we did in the last lesson. Students should be able to work through questions #1-4. For the work they are showing, we are expecting that they are showing their computation but not function notation. We will add that later. As you are checking in with groups, we want to get them thinking about the functions they are using and what the function notation might look like. Use these guiding questions to get students ready for this next level of formality.

Guiding Questions

- What are you plugging in to the function? What are you getting out?
- Where did you get that input? How did you find it?
- Is it possible to skip this whole middle column with the number of pies?
- Could we make a table that only showed the number of hours and the profit made?

You may expect students to find going from the table to writing a function that inputs hours and outputs profit to be a bit of a jump, but you may be surprised at how well they do with it. Let them leave their function unsimplified if they want. This helps them to see each function in the composition and to understand how an **entire function** is being input.

#### Formalize Later

Before teaching this lesson, we'd suggest you plan out your debrief, including when you will layer on the function notation and in what order. When creating this answer key, we used lots of different colors to help students see which functions and variables were being used for each question. We would recommend using different colors in this way for your debrief also.

The order in which you add the margin notes matters also. For this debrief, we would start by having students share their thinking as you go over the work as a class. If you need them to elaborate, use the Guiding Questions above. After they explain, we want to formalize their computations with function notation using the different colors. We used green, purple, and orange for this on the answer key. As you write out the function notation, ask the student again what they plugged in and what they got out. Follow this same process for all of the margin notes that are shown in green, purple and orange. These notes focus on defining the functions individually.

When we get to debriefing question #5, now we want to focus on the composition of functions. We wrote the margin notes about this in red on the answer key. Students evaluated the composition of functions using the table. We need to layer that with function notation now. Go back to question #3 and point out to students that in part a, we found the number of pies made in 5 hours, which is *N*(5)=40. Then we took that answer and plugged it into the profit function, which is *P*(40)=535. Or another way of looking at it is *P*(*N*(5))=535. Add this into the margin. Now we can look at the function they wrote for #5 and can identify that it is *P*(*N*(*t*)) which is a composition of functions!