Is the Inverse a Function? Mastering the Horizontal Line Test

PreAlgebra Grades High School 1:22 Video
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Lesson Description

This lesson explores the horizontal line test, a crucial tool in precalculus for determining whether the inverse of a function is also a function. Through visual examples and clear explanations, students will learn how to apply this test and understand its implications for function invertibility.

Video Resource

What is the Horizontal Line Test?

Mario's Math Tutoring

Duration: 1:22
Watch on YouTube

Key Concepts

  • Function Invertibility
  • Horizontal Line Test
  • Relationship between a function and its inverse

Learning Objectives

  • Students will be able to apply the horizontal line test to a given graph to determine if its inverse is a function.
  • Students will be able to explain the relationship between the horizontal line test and the invertibility of a function.
  • Students will be able to connect the horizontal line test to the vertical line test and understand what they assess.

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 their graphical representation (reflection over y=x).
  • Video Presentation (3 mins)
    Play the 'What is the Horizontal Line Test?' video from Mario's Math Tutoring. Instruct students to pay close attention to the examples provided.
  • Guided Practice (10 mins)
    Work through the examples from the video again, pausing to ask students questions about why the horizontal line test passes or fails in each case. Emphasize the connection between the number of intersection points and the invertibility of the function.
  • Independent Practice (10 mins)
    Provide students with a worksheet containing various graphs. Have them apply the horizontal line test to each graph and determine whether the inverse is a function. Include examples of polynomial, rational, and trigonometric functions.
  • Discussion and Wrap-up (7 mins)
    Discuss the results of the independent practice. Address any common misconceptions. Reiterate the key concept: The horizontal line test determines if the *inverse* of a function is a function. Tie it to the vertical line test showing when the *original* function is a function.

Interactive Exercises

  • Graphing Tool Activity
    Use an online graphing tool (e.g., Desmos, GeoGebra) to plot functions. Students can then visually apply the horizontal line test and confirm their findings by examining the inverse function's graph.

Discussion Questions

  • Why does the horizontal line test work to determine if the inverse of a function is a function?
  • What does it mean if a graph passes the vertical line test but fails the horizontal line test?

Skills Developed

  • Visual Analysis
  • Critical Thinking
  • Function Analysis

Multiple Choice Questions

Question 1:

The horizontal line test is used to determine if:

Correct Answer: The inverse of a function is a function.

Question 2:

If a graph intersects a horizontal line at more than one point, the inverse of the function is:

Correct Answer: Not a function.

Question 3:

Which of the following functions will NOT have an inverse that is also a function?

Correct Answer: f(x) = x^2

Question 4:

The horizontal line test is graphically assessing whether a function is:

Correct Answer: One-to-one

Question 5:

What is the range of the inverse function equal to?

Correct Answer: The domain of the original function.

Question 6:

A function passes the vertical line test but fails the horizontal line test. What does this imply?

Correct Answer: The function is a function, but its inverse is not a function.

Question 7:

Which of the following transformations must be applied to a function for a horizontal line test?

Correct Answer: No Transformation

Question 8:

If the inverse function has a domain of all real numbers, then original function's ____ is also all real numbers.

Correct Answer: Range

Question 9:

The function f(x) = sin(x) fails the horizontal line test. How can we restrict its domain to ensure its inverse is a function?

Correct Answer: [-π/2, π/2]

Question 10:

Which function fails the Horizontal Line Test for all Real Numbers?

Correct Answer: y = |x|

Fill in the Blank Questions

Question 1:

The horizontal line test determines if the _________ of a function is a function.

Correct Answer: inverse

Question 2:

If a function passes both the vertical and horizontal line tests, it is considered a ___________ function.

Correct Answer: one-to-one

Question 3:

If a horizontal line intersects a graph at three points, then the inverse is _________ a function.

Correct Answer: not

Question 4:

For a function to have an inverse that's also a function, it must be ________ over its entire domain.

Correct Answer: one-to-one

Question 5:

The graph of the inverse function is a _________ of the original function across the line y=x.

Correct Answer: reflection

Question 6:

The ________ line test checks if the original relation is a function.

Correct Answer: vertical

Question 7:

A function that is strictly increasing or strictly decreasing will always _________ the horizontal line test.

Correct Answer: pass

Question 8:

When restricting the domain of a function to create an invertible function, we must ensure that new function is ______.

Correct Answer: one-to-one

Question 9:

The domain of an inverse function will be the __________ of the original function.

Correct Answer: range

Question 10:

The inverse of the function f(x) = x is f⁻¹(x) = _________.

Correct Answer: x

Educational Standards

PC.F.1.5:Determine if a function has an inverse. Algebraically and graphically find the inverse or define any restrictions on the domain that meet the requirement for invertibility, and find the inverse on the restricted domain. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.1.3:Interpret characteristics of graphs and tables for a function that models a relationship between two quantities in terms of the quantities.

Teaching Materials

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