Restricting Domains for Invertible Functions: Mastering the Horizontal Line Test

Algebra 2 Grades High School 12:10 Video
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Lesson Description

Explore how to restrict the domain of a function to ensure its inverse is also a function. Learn about the vertical and horizontal line tests, domain and range, and graphical representation of inverse functions.

Video Resource

How to Restrict the Domain so the Inverse is a Function

Mario's Math Tutoring

Duration: 12:10
Watch on YouTube

Key Concepts

  • Vertical Line Test
  • Horizontal Line Test
  • Domain and Range
  • Inverse Functions
  • Restricting the Domain

Learning Objectives

  • Students will be able to determine if a function has an inverse that is also a function using the horizontal line test.
  • Students will be able to restrict the domain of a function to make its inverse a function.
  • Students will be able to identify the domain and range of a function and its inverse.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a function and the vertical line test. Briefly discuss the concept of inverse functions and why some functions do not have inverses that are also functions.
  • Video Presentation (15 mins)
    Play the video 'How to Restrict the Domain so the Inverse is a Function' by Mario's Math Tutoring. Encourage students to take notes on key concepts and examples.
  • Horizontal Line Test (10 mins)
    Explain the horizontal line test and its relationship to the existence of an inverse function. Provide examples of functions that pass and fail the horizontal line test.
  • Restricting the Domain (15 mins)
    Demonstrate how to restrict the domain of a function to make it pass the horizontal line test. Work through examples, emphasizing the graphical representation of the restricted function and its inverse.
  • Practice Problems (15 mins)
    Assign practice problems where students must determine if a function has an inverse, and if not, restrict the domain to create an invertible function.

Interactive Exercises

  • Graphing Activity
    Students graph various functions on Desmos or a graphing calculator and use the horizontal line test to determine if an inverse function exists. If it doesn't, they restrict the domain graphically and algebraically.
  • Think-Pair-Share
    Present a function and have students individually determine if it has an inverse. Then, they pair up to discuss their findings and justify their answers. Finally, share results and reasoning with the whole class.

Discussion Questions

  • Why is it important for an inverse to also be a function?
  • How does restricting the domain change the graph of a function?
  • Can you always restrict the domain of a function to make it invertible? Why or why not?

Skills Developed

  • Analytical Thinking
  • Problem-Solving
  • Graphical Interpretation
  • Algebraic Manipulation

Multiple Choice Questions

Question 1:

Which test is used to determine if the inverse of a function is also a function?

Correct Answer: Horizontal Line Test

Question 2:

What does it mean to restrict the domain of a function?

Correct Answer: To limit the possible input values of the function

Question 3:

If a function fails the horizontal line test, what does this indicate about its inverse?

Correct Answer: The inverse is not a function.

Question 4:

Consider the function f(x) = x². To make its inverse a function, which domain restriction is most appropriate?

Correct Answer: x ≥ 0

Question 5:

What is the range of the inverse function if the domain of the original restricted function is x ≥ 2?

Correct Answer: y ≥ 2

Question 6:

Which of the following functions requires a domain restriction for its inverse to be a function?

Correct Answer: f(x) = x² - 4

Question 7:

When restricting the domain of a parabola, where is the best place to cut the parabola?

Correct Answer: At its vertex

Question 8:

What happens to the domain of f(x) when finding its inverse f⁻¹(x)?

Correct Answer: It becomes the range of f⁻¹(x)

Question 9:

What are the x-values called?

Correct Answer: Domain

Question 10:

What are the y-values called?

Correct Answer: Range

Fill in the Blank Questions

Question 1:

The ___________ Line Test is used to determine if a function has an inverse function.

Correct Answer: Horizontal

Question 2:

If a function fails the Horizontal Line Test, the ___________ must be restricted so that its inverse is also a function.

Correct Answer: domain

Question 3:

If f(x) = x², restricting the domain to x ≥ 0 means x can only be __________ numbers.

Correct Answer: positive

Question 4:

When you switch the x and y-coordinates of a function, it reflects across the line ___________.

Correct Answer: y=x

Question 5:

The _______ of the original function becomes the domain of the inverse function.

Correct Answer: range

Question 6:

Restricting the domain often involves finding the ________ of a quadratic function.

Correct Answer: vertex

Question 7:

A function must be _________ to have an inverse that is also a function without any domain restrictions.

Correct Answer: one-to-one

Question 8:

Functions that pass the vertical line test are called __________.

Correct Answer: functions

Question 9:

The y-values are called the ___________.

Correct Answer: range

Question 10:

The x-values are called the __________.

Correct Answer: domain

Educational Standards

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.F.1:Understand functions as descriptions of covariation (how related quantities vary together).

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