# 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 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**

- How many tiles would be needed if the length of the pool was 18 feet? 21 feet?
- Can you write a rule using _______'s method for counting the tiles?
- What's the same about these different counting methods? What's different?
- What is the input of this function? What is the output?
- How does the side length of the pool affect the cost?

#### 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.