Unraveling Composite Functions: A Decomposition Expedition

PreAlgebra Grades High School 6:56 Video
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Lesson Description

Explore the reverse process of function composition by decomposing composite functions into their original components. Learn to identify inner and outer functions and verify your solutions.

Video Resource

How to Decompose a Composite Function

Mario's Math Tutoring

Duration: 6:56
Watch on YouTube

Key Concepts

  • Composite Functions
  • Decomposition of Functions
  • Inner and Outer Functions

Learning Objectives

  • Students will be able to identify the inner and outer functions of a composite function.
  • Students will be able to decompose a given composite function into two or more possible original functions.
  • Students will be able to verify the decomposition by composing the resulting functions and comparing them to the original composite function.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of composite functions: f(g(x)). Explain that this lesson will cover the reverse process: starting with h(x) and finding f(x) and g(x) such that h(x) = f(g(x)).
  • Video Viewing (15 mins)
    Play the YouTube video "How to Decompose a Composite Function" by Mario's Math Tutoring. Encourage students to take notes on the methods presented for identifying inner and outer functions.
  • Guided Practice (20 mins)
    Work through the examples from the video on the board, emphasizing the thought process behind choosing the inner and outer functions. Discuss the importance of verifying the decomposition by composing the functions. Present additional examples beyond the video, gradually increasing in complexity.
  • Independent Practice (15 mins)
    Provide students with a set of composite functions to decompose on their own. Encourage them to explore different possibilities and verify their solutions. Circulate to provide assistance and answer questions.
  • Wrap-up and Discussion (5 mins)
    Summarize the key concepts of the lesson and address any remaining questions. Preview upcoming topics related to function composition and inverses.

Interactive Exercises

  • Decomposition Challenge
    Present a complex composite function and have students work in small groups to find as many different decompositions as possible within a set time limit. Groups then share their solutions and discuss the reasoning behind their choices.

Discussion Questions

  • Why can a composite function often be decomposed in multiple ways?
  • How does understanding the order of operations help in decomposing a function?
  • How can domain restrictions affect the possible decompositions of a function?

Skills Developed

  • Analytical Thinking
  • Problem Solving
  • Function Manipulation

Multiple Choice Questions

Question 1:

Which of the following is a possible decomposition of h(x) = (x^2 + 1)^3, where f(x) is the outer function and g(x) is the inner function?

Correct Answer: f(x) = x^3, g(x) = x^2 + 1

Question 2:

If h(x) = √ (2x - 5), which of the following could be the inner function g(x)?

Correct Answer: 2x - 5

Question 3:

Given h(x) = 1/(x + 3)^2, if g(x) = x + 3, what is the corresponding outer function f(x)?

Correct Answer: 1/x^2

Question 4:

Which statement best describes the process of decomposing a composite function?

Correct Answer: Finding two functions that, when composed, equal the original function.

Question 5:

Why is it important to verify your decomposition of a composite function?

Correct Answer: To ensure the composition of your decomposed functions equals the original function

Question 6:

For h(x) = sin(x^2), which of the following represents a correct decomposition?

Correct Answer: f(x) = sin(x), g(x) = x^2

Question 7:

If h(x) = (x - 4)^5 + 2, and g(x) = (x - 4)^5, what would f(x) be?

Correct Answer: x + 2

Question 8:

Which of the following is NOT a valid way to think about decomposing a composite function?

Correct Answer: Simplifying the original function as much as possible

Question 9:

Given h(x) = |x + 1|, what could g(x) be?

Correct Answer: x + 1

Question 10:

What is the first step in decomposing h(x)?

Correct Answer: Identifying potential inner and outer functions

Fill in the Blank Questions

Question 1:

Decomposing a composite function is the _________ of composing functions.

Correct Answer: reverse

Question 2:

In the composition f(g(x)), g(x) is referred to as the _________ function.

Correct Answer: inner

Question 3:

In the composition f(g(x)), f(x) is referred to as the _________ function.

Correct Answer: outer

Question 4:

To verify a decomposition, you should _______ the functions you find for f(x) and g(x).

Correct Answer: compose

Question 5:

If h(x) = (5x)^2, a possible inner function g(x) is _________.

Correct Answer: 5x

Question 6:

If h(x) = (5x)^2, and the inner function is 5x, then the outer function, f(x) is _________.

Correct Answer: x^2

Question 7:

A composite function can sometimes have _________ than one possible decomposition.

Correct Answer: more

Question 8:

When decomposing a function, consider what is being plugged _________ another function.

Correct Answer: into

Question 9:

Before settling on a decomposition, _________ that f(g(x)) = h(x).

Correct Answer: verify

Question 10:

Finding the inner and outer functions is the core of function _________.

Correct Answer: decomposition

Educational Standards

PC.F.2.2:[Objective] Rewrite a function as a composition of functions. PC.F.2.1:[Objective] Model relationships through composition, and attend to the restrictions of the domain.

Teaching Materials

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