Compositions of Functions (Lesson 1.8)
Unit 0: Prerequisites
Day 1: The Cartesian Plane
Day 2: Equations of Circles
Day 3: Solving Equations in Multiple Representations
Day 4: Reasoning with Formulas
Day 5: Quiz 0.1 to 0.4
Day 6: Linear Relationships
Day 7: Reasoning with Slope
Day 8: Set Notation
Day 9: Quiz 0.5 to 0.7
Day 10: Unit 0 Review
Day 11: Unit 0 Test
Unit 1: Functions
Day 1: Functions and Function Notation
Day 2: Domain and Range
Day 3: Rates of Change and Graph Behavior
Day 4: Library of Parent Functions
Day 5: Transformations of Functions
Day 6: Transformations of Functions
Day 7: Even and Odd Functions
Day 8: Quiz 1.1 to 1.6
Day 9: Building Functions
Day 10: Compositions of Functions
Day 11: Inverse Functions
Day 12: Graphs of Inverse Functions
Day 13: Piecewise Functions
Day 14: Quiz 1.7 to 1.11
Day 15: Unit 1 Review
Day 16: Unit 1 Test
Unit 2: Polynomial and Rational Functions
Day 1: Connecting Quadratics
Day 2: Completing the Square
Day 3: Polynomials in the Short Run
Day 4: Polynomials in the Long Run
Day 5: Review 2.1-2.4
Day 6: Quiz 2.1 to 2.4
Day 7: Factor and Remainder Theorem
Day 8: Factor and Remainder Theorem
Day 9: Complex Zeros
Day 10: Connecting Zeros Across Multiple Representations
Day 11: Intro to Rational Functions
Day 12: Graphing Rational Functions
Day 13: Quiz 2.5 to 2.9
Day 14: Unit 2 Review
Day 15: Unit 2 Test
Unit 2: Linear Algebra
Unit 3: Exponential and Logarithmic Functions
Day 1: Exponential Functions
Day 2: Graphs of Exponential Functions
Day 3: Compound Interest and an Introduction to "e"
Day 4: Review 3.1-3.3
Day 5: Quiz 3.1 to 3.3
Day 6: Logarithmic Functions
Day 7: Graphs of Logarithmic Functions
Day 8: Logarithm Properties
Day 9: Solving Exponential and Logarithmic Equations
Day 10: Quiz 3.4 to 3.7
Day 11: Exponential and Logarithmic Modeling
Day 12: Unit 3 Review
Day 13: Unit 3 Test
Unit 4: Trigonometric Functions
Day 1: Right Triangle Trig
Day 2: Inverse Trig Ratios
Day 3: Radians and Degrees
Day 4: Unit Circle
Day 5: Unit Circle
Day 6: Other Trig Functions
Day 7: Review 4.1-4.6
Day 8: Quiz 4.1 to 4.6
Day 9: Graphing Sine and Cosine
Day 10: Transformations of Sine and Cosine Graphs
Day 11: Graphing Secant and Cosecant
Day 12: Graphing Tangent and Cotangent
Day 13: Quiz 4.7 to 4.10
Day 14: Inverse Trig Functions
Day 15: Trigonometric Modeling
Day 16: Trigonometric Identities
Day 17: Unit 4 Review
Day 18: Unit 4 Review
Day 19: Unit 4 Test
Unit 5: Applications of Trigonometry
Day 1: Law of Sines
Day 2: The Ambiguous Case (SSA)
Day 3: Law of Cosines
Day 4: Area and Applications of Laws
Day 5: Vectors
Day 6: Review 5.1-5.5
Day 7: Quiz 5.1 to 5.5
Day 8: Polar Coordinates
Day 9: Equations in Polar and Cartesian Form
Day 10: Polar Graphs Part 1
Day 11: Polar Graphs Part 2
Day 12: Review 5.6-5.9
Day 13: Quiz 5.6 to 5.9
Day 14: Parametric Equations
Day 15: Parametric Equations (With Trig)
Day 16: Unit 5 Review
Day 17: Unit 5 Test
Unit 6: Systems of Equations
Day 1: What is a Solution?
Day 2: Solving Systems with Substitution
Day 3: Solving Systems with Elimination
Day 4: Review 6.1-6.3
Day 5: Quiz 6.1 to 6.3
Day 6: Solving Systems in 3 Variables
Day 7: Solving Systems in 3 Variables
Day 8: Partial Fractions
Day 9: Unit 6 Review
Day 10: Unit 6 Test
Unit 7: Sequences and Series
Day 1: Introducing Sequences
Day 2: Using Sequences and Series to Describe Patterns
Day 3: Arithmetic Sequences and Series
Day 4: Review 7.1-7.2
Day 5: Quiz 7.1 to 7.2
Day 6: Geometric Sequences and Finite Series
Day 7: Infinite Geometric Sequences and Series
Day 8: Proof by Induction
Day 9: Proof by Induction
Day 10: Quiz 7.3 to 7.5
Day 11: Unit 7 Review
Day 12: Unit 7 Test
Unit 8: Limits
Day 1: What is a Limit?
Day 2: Evaluating Limits Graphically
Day 3: Evaluating Limits with Direct Substitution
Day 4: Evaluating Limits Analytically
Day 5: Evaluating Limits Analytically
Day 6: Review 8.1-8.4
Day 7: Quiz 8.1 to 8.4
Day 8: Continuity
Day 9: Continuity
Day 10: Intermediate Value Theorem
Day 11: Intermediate Value Theorem
Day 12: Review 8.5-8.6
Day 13: Quiz 8.5 to 8.6
Day 14: Limits at Infinity
Day 15: Unit 8 Review
Day 16: Unit 8 Test
Unit 9: Derivatives
Day 1: Introduction to Derivatives
Day 2: Average versus Instantaneous Rates of Change
Day 3: Calculating Instantaneous Rate of Change
Day 4: Calculating Instantaneous Rate of Change
Day 5: The Derivative Function
Day 6: The Derivative Function
Day 7: Review 9.1-9.3
Day 8: Quiz 9.1 to 9.3
Day 9: Derivative Shortcuts
Day 10: Differentiability
Day 11: Connecting f and f’
Day 12: Connecting f and f’
Day 13: Review 9.4-9.6
Day 14: Quiz 9.4 to 9.6
Day 15: Derivatives of Sine and Cosine
Day 16: Product Rule
Day 17: Quotient Rule
Day 18: Review 9.7-9.9
Day 19: Quiz 9.7 to 9.9
Day 20: Unit 9 Review
Day 21: Unit 9 Test
Unit 10: (Optional) Conic Sections
Day 1: Intro to Conic Sections
Day 2: Defining Parabolas
Day 3: Working with Parabolas
Day 4: Quiz 10.1 to 10.3
Day 5: Defining Ellipses
Day 6: Working with Elllipses
Day 7: Defining Hyperbolas
Day 8: Working with Hyperbolas
Day 9: Quiz 10.4 to 10.7
Day 10: Unit 10 Review
Day 11: Unit 10 Test
Learning Targets
Understand that when two functions are composed, the output of one function becomes the input of the other
Interpret the input and output of composite functions in context
Evaluate and write equations for compositions of functions by plugging the inner function in as the independent variable of the outer function
Find the domain of a composition of functions
Tasks/Activity | Time |
---|---|
Activity | 20 minutes |
Debrief Activity | 10 minutes |
Important Ideas | 5 minutes |
Check Your Understanding | 15 minutes |
Activity: How Much Does it Cost to Tile a Pool?
Lesson Handouts
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Answer Key
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Homework
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Experience First
In this lesson students build their own composite function by expressing regularity in repeated reasoning. First, students consider how the length of a pool determines the number of tiles that are needed to make a border for the pool. Algebraically there are many ways to come up with this expression, so encourage students to use color to demonstrate how they “see” the tiles being added. This is a great opportunity to talk about the equivalence of expressions!
After determining the number of tiles, students go on to figure out the cost of those tiles with the included delivery fee. As you monitor groups, ask students questions like “what determines the cost of the project?” or “how/why would increasing the length of the pool affect the cost?” Students should articulate that the number of tiles determines the cost, but the length of the pool determines the number of tiles. Be listening for phrases like “increasing the length of the pool increases the number of tiles, which then increases the cost of the project”. This kind of sequential reasoning is critical for developing the students’ understanding of composite functions.
When completing the table, it will be helpful if students show their work for calculating the number of tiles and cost of the project. When students see 2(18)+16=52 and in the next column 52(5.75)+9.99, it becomes evident how the output of the first function becomes the input of the second function. Finally we want students to see how this can be stated in one equation, namely by inserting the expression 2x+16 into the cost equation to represent the number of tiles (as determined by the length of the pool). Throughout the experience students are asked to attend to the kinds of values that go into a function, and those that come out. Restricted domains for the length of the pool creates a restricted range for the number of tiles; which ultimately determines the price range to complete the project. Guiding Questions
Formalize Later
As always, a lot of formal notation is omitted in the experience and then layered on during the formalization. Support students to see how C(n(x)) demonstrates the sequence of equations and the inputs and outputs of each “stage”. We use letters that represent the context instead of the traditional f(g(x)).
Understanding of composite functions is critical for success in AP Calculus. Students must be able to work flexibly with composite functions represented numerically, graphically, or analytically. Consider assigning homework problems that mix representations such as question 3 of the Check Your Understanding (graphical and analytical). Finding the domain of a composite function tends to be the hardest part of the lesson for students. You may wish to spend some time going over the second question of the Check Your Understanding. Make sure that students know that it’s not enough to consider only the domain of the inside function, but that they must think about what outputs are produced by those inputs and if those can be inserted into the output function.