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 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!