Use the composition of two functions to model and solve real-world problems

Standards Alignment

• TEKS.111.42.02A primary
  use the composition of two functions to model and solve real-world problems;
• TEKS.111.42.02B primary
  demonstrate that function composition is not always commutative;
• TEKS.111.42.02C primary
  represent a given function as a composite function of two or more functions;

Lesson Overview

This lesson introduces Grade 12 students to the concept of composing two functions to model and solve real-world problems. Students will explore how to represent functions as composites, understand that composition is not commutative, and apply these ideas to practical situations. The lesson is aligned with TEKS standards for Precalculus and emphasizes multiple representations and algebraic construction of functions.

Learning Objectives

• Use the composition of two functions to model and solve real-world problems.
• Demonstrate that function composition is not always commutative.
• Represent a given function as a composite function of two or more functions.

Success Criteria

• Students can compose two functions to create a new function that models a real-world scenario.
• Students can explain and show with examples that composing functions in different orders gives different results.
• Students can break down a function into two or more component functions and write it as a composite function.

Prerequisite Knowledge

Students should understand the definition of a function, how to evaluate functions, and basic algebraic manipulation of expressions.

Key Vocabulary

• Function
• Composition of functions
• Composite function
• Commutative property
• Input
• Output
• Domain
• Range

Materials and Resources

• Graphing calculator or graphing software
• Whiteboard and markers
• Paper and pencils
• Real-world problem scenarios (printed or projected)

Teacher Preparation

• Prepare examples of real-world problems that can be modeled using function composition.
• Prepare guided practice problems illustrating composition and non-commutativity.
• Have graphing tools ready for demonstrations.
• Prepare a visual diagram showing function composition process.

Detailed Lesson Notes

Definition of Function Composition

Function composition means applying one function to the result of another function. If we have two functions f and g, the composition f(g(x)) means first apply g to x, then apply f to the result. This creates a new function that combines the effects of both.

Modeling Real-World Problems Using Composition

Many real-world situations involve multiple steps or processes. Each step can be represented by a function. Composing these functions models the entire process. For example, if g(x) represents the cost of materials based on quantity x, and f(y) represents the total price including tax based on cost y, then f(g(x)) models the total price based on quantity.

Non-Commutativity of Function Composition

Unlike addition or multiplication, composing functions is generally not commutative. That is, f(g(x)) is usually not equal to g(f(x)). This means the order in which functions are composed matters. Teachers should provide examples where switching the order changes the output to illustrate this clearly.

Representing Functions as Composite Functions

Sometimes a complex function can be broken down into simpler functions composed together. For example, if h(x) = (2x + 3)^2, we can write h as the composition of f(x) = x^2 and g(x) = 2x + 3, so h(x) = f(g(x)). This helps in understanding and analyzing functions by their components.

Worked Example: Modeling a Real-World Problem

Suppose a company charges a base fee plus a rate per hour for a service. Let g(x) = 50 + 20x represent the total cost before tax for x hours. The tax function is f(y) = 1.08y (an 8% tax). The total cost including tax is f(g(x)) = 1.08(50 + 20x). This composition models the final price based on hours used.

Worked Examples

Classroom Activity

Students calculate and compare f(g(x)) and g(f(x)) for given x values, noting differences.

Guided Practice

Independent Practice

1. Foundational: If f(x) = 2x and g(x) = x + 3, what is f(g(5))?
2. Developing: Given f(x) = x^2 and g(x) = x - 1, calculate g(f(3)) and f(g(3)). Are they equal?
3. Application: Write the function h(x) = (4x + 1)^3 as a composition of two functions f and g.
4. Analysis: Explain why function composition is not commutative using the functions f(x) = 2x and g(x) = x + 5.
5. Challenge: A company charges a base fee of $40 plus $15 per hour for a service. Tax is applied at 10% on the total cost. Define functions to model the total cost including tax for h hours and write the composite function. Then calculate the total cost for 3 hours.

Independent Practice Teacher Answer Key

1. 1. g(5) = 5 + 3 = 8; then f(g(5)) = f(8) = 2 * 8 = 16.
2. 2. f(3) = 3^2 = 9; g(f(3)) = g(9) = 9 - 1 = 8. g(3) = 3 - 1 = 2; f(g(3)) = f(2) = 2^2 = 4. Since 8 ≠ 4, they are not equal.
3. 3. Let g(x) = 4x + 1 and f(x) = x^3. Then h(x) = f(g(x)).
4. 4. f(g(x)) = 2(x + 5) = 2x + 10, while g(f(x)) = (2x) + 5 = 2x + 5. Since 2x + 10 ≠ 2x + 5, the compositions differ, showing that order matters and composition is not commutative.
5. 5. Let g(h) = 40 + 15h (cost before tax), and f(c) = 1.10c (cost including tax). The composite function is f(g(h)) = 1.10(40 + 15h). For h = 3, g(3) = 40 + 15*3 = 85; f(g(3)) = 1.10 * 85 = 93.5. Total cost is $93.50.

Guiding Questions

• What does it mean to compose two functions?
• Can you show an example where f(g(x)) is not equal to g(f(x))?
• How can you break down a function into two functions composed together?
• Why is function composition useful in modeling real-world problems?

Common Misconceptions

• Students may think function composition is commutative and expect f(g(x)) to equal g(f(x)).
• Students might confuse the order of applying functions in composition.
• Students may struggle to identify inner and outer functions when decomposing a composite function.

Differentiation

Assessment

• Have students solve function composition problems with given functions.
• Ask students to explain why f(g(x)) may differ from g(f(x)) with examples.
• Check students' ability to write functions as compositions of two functions.
• Write a composite function for h(x) = (3x - 2)^4 and identify the inner and outer functions.
• Calculate f(g(2)) and g(f(2)) for f(x) = x + 1 and g(x) = 2x and explain if they are equal.
• Model a real-world problem using function composition and solve for a given input.
• Explain with examples why function composition is not commutative.
• Decompose a complex function into a composite of simpler functions and verify by substitution.

Answer Guide

• Calculate f(g(4)) and g(f(4)) for f(x) = 3x + 2 and g(x) = x^2.
  Answer: f(g(4)) = 50 and g(f(4)) = 196; they are not equal, showing composition is not commutative.
• Write h(x) = (4x + 1)^3 as a composite function.
  Answer: h(x) = f(g(x)) where g(x) = 4x + 1 and f(x) = x^3.
• Explain why function composition is not commutative using f(x) = 2x and g(x) = x + 5.
  Answer: f(g(x)) = 2x + 10 and g(f(x)) = 2x + 5; since these are different, composition order matters.
• Model total cost including tax for a service with base fee and hourly rate using composition.
  Answer: Define g(h) = 40 + 15h and f(c) = 1.10c; total cost is f(g(h)) = 1.10(40 + 15h).
• Calculate total cost including tax for 3 hours using the composite function.
  Answer: Total cost = 1.10(40 + 15*3) = 1.10 * 85 = $93.50.

Real-Life Application

Ask students to find a real-world example at home or in their community where two-step calculations occur, such as cooking or budgeting, and describe how function composition could model that process.

Homework or Home Connection

• Ask students to find a real-world example at home or in their community where two-step calculations occur, such as cooking or budgeting, and describe how function composition could model that process.

Lesson Summary

In this lesson, students learned how to compose two functions to model and solve real-world problems. They explored that function composition is generally not commutative, meaning the order of composition affects the result. Students practiced representing functions as composites of simpler functions and applied these concepts to practical scenarios, reinforcing their understanding of function composition in algebra and real life.

Teacher Notes

Use the exact standards alignment and retrieved-source provenance stored with this enrichment.

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