Function Inverses: Verification Through Composition

Algebra 2 Grades High School 3:29 Video
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Lesson Description

This lesson explores how to algebraically verify if two functions are inverses of each other using function composition. We'll cover the definition of inverse functions, the process of finding an inverse, and the method of composing functions to prove their inverse relationship.

Video Resource

Verifying Functions are Inverses

Mario's Math Tutoring

Duration: 3:29
Watch on YouTube

Key Concepts

  • Inverse Functions
  • Function Composition
  • Algebraic Verification

Learning Objectives

  • Students will be able to find the inverse of a given function.
  • Students will be able to verify if two functions are inverses of each other through composition.
  • Students will be able to explain the relationship between a function and its inverse.

Educator Instructions

  • Introduction (5 mins)
    Begin by discussing the concept of inverse operations (addition/subtraction, multiplication/division). Then, introduce the idea of inverse functions as operations that "undo" each other. Briefly explain the notation for inverse functions.
  • Video Presentation (10 mins)
    Play the video "Verifying Functions are Inverses" by Mario's Math Tutoring. Encourage students to take notes on the key steps and examples.
  • Example 1 Walkthrough (10 mins)
    Reiterate the first example from the video (f(x) = 2x - 7). Guide students through finding the inverse and then verifying the inverse relationship using function composition (both f(f⁻¹(x)) and f⁻¹(f(x))).
  • Example 2 Walkthrough (10 mins)
    Reiterate the second example from the video (f(x) = (3x + 5)/2). Guide students through finding the inverse and then verifying the inverse relationship using function composition (both f(f⁻¹(x)) and f⁻¹(f(x))).
  • Independent Practice (15 mins)
    Provide students with practice problems where they need to find the inverse of a function and verify if two given functions are inverses. Offer assistance as needed.
  • Wrap-up and Q&A (5 mins)
    Summarize the key concepts and address any remaining questions from the students.

Interactive Exercises

  • Find the Inverse and Verify
    Give students pairs of functions. For each pair, they must determine if the functions are inverses by first finding the inverse of one function and then using composition to verify. Functions include: a) f(x) = x/3 + 2, g(x) = 3x - 6 b) f(x) = x^3 - 1, g(x) = ∛(x+1) c) f(x) = (x-4)/5, g(x) = 5x + 4

Discussion Questions

  • What does it mean for two functions to be inverses of each other?
  • Why is it necessary to check function composition in both orders (f(g(x)) and g(f(x))) to verify inverse functions?
  • Can you think of real-world examples of inverse operations or processes?

Skills Developed

  • Algebraic Manipulation
  • Function Composition
  • Problem-Solving

Multiple Choice Questions

Question 1:

What is the primary method for verifying that two functions, f(x) and g(x), are inverses of each other?

Correct Answer: Using function composition: f(g(x)) = x and g(f(x)) = x

Question 2:

If f(x) = x + 5, what is its inverse function, f⁻¹(x)?

Correct Answer: f⁻¹(x) = x - 5

Question 3:

If f(g(x)) = x and g(f(x)) = x, what can you conclude about f(x) and g(x)?

Correct Answer: They are inverse functions

Question 4:

Which of the following pairs of functions are NOT inverses of each other?

Correct Answer: f(x) = x^2, g(x) = √x

Question 5:

What is the first step in finding the inverse of a function y = f(x)?

Correct Answer: Switch x and y

Question 6:

Suppose f(x) = 3x + 2 and g(x) = (x - 2)/3. What is f(g(x))?

Correct Answer: x

Question 7:

Why is it important to verify inverse functions using both f(g(x)) and g(f(x))?

Correct Answer: To ensure the relationship holds regardless of the order of application

Question 8:

If f(x) = √x, what is the domain restriction needed for its inverse, f⁻¹(x) = x² to be a true inverse?

Correct Answer: x ≥ 0

Question 9:

What does the notation f⁻¹(x) represent?

Correct Answer: The inverse of f(x)

Question 10:

Given f(x) = (x-1)/4, find f⁻¹(5).

Correct Answer: 21

Fill in the Blank Questions

Question 1:

To find the inverse of a function, the first step is to _______ x and y.

Correct Answer: switch

Question 2:

If f(x) = 4x, then f⁻¹(x) = _______.

Correct Answer: x/4

Question 3:

The composition f(g(x)) represents plugging the function _______ into the function f(x).

Correct Answer: g(x)

Question 4:

If f(x) and g(x) are inverses, then f(g(x)) = _______.

Correct Answer: x

Question 5:

The inverse of a function 'undoes' the _______ of the original function.

Correct Answer: operation

Question 6:

Before finding the inverse of f(x) = √x, the domain restriction is usually defined as x _______ 0.

Correct Answer: ≥

Question 7:

If f(x) = x - 3, then f⁻¹(x) = _______.

Correct Answer: x+3

Question 8:

The notation for the inverse of f(x) is written as _______.

Correct Answer: f⁻¹(x)

Question 9:

When verifying inverse functions, you must check both f(g(x)) and _______.

Correct Answer: g(f(x))

Question 10:

If f(x) = 2x - 1, then f⁻¹(x) = _______.

Correct Answer: (x+1)/2

Educational Standards

A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together). A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain. A2.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

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