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Horizontal Line Test: Definition & Overview

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  • 0:00 One-To-One Function Defined
  • 1:00 The Horizontal Line Test
  • 1:15 How It Works?
  • 2:00 Why Is It Important?
  • 2:55 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
The horizontal line test is a test used to determine if a function is a one-to-one function. This lesson will define how to perform the horizontal line test as well as give some examples.

One-to-One Function Defined

The horizontal line test is a method that can be used to determine if a function is a one-to-one function. This means that, for every y-value in the function, there is only one unique x-value. One simple example of a one-to-one function (often called an injective function) is with the daily specials at a restaurant. Let's say the specials are as follows:

  • Sunday - Meat Loaf
  • Monday - Turkey
  • Tuesday - Fried Chicken
  • Wednesday - Steak
  • Thursday - Pork Chops
  • Friday - Salmon
  • Saturday - Pot Roast

This example represents a one-to-one function because, for every y-value (day of the week) there is only one unique x-value (dinner special).

A non-example might be with these breakfast specials at the same restaurant:

  • Sunday - Omelet
  • Monday - Pancakes
  • Tuesday - Scrambled Eggs
  • Wednesday - Waffles
  • Thursday - Scrambled Eggs
  • Friday - French Toast
  • Saturday - Eggs Benedict

This is not an injective function because, for two days (Tuesday and Thursday), the special is the same -scrambled eggs.

The Horizontal Line Test

The horizontal line test is a method to determine if a function is a one-to-one function or not. It is used exclusively on functions that have been graphed on the coordinate plane. It is called the horizontal line test because the test is performed using a horizontal line, which is a line that runs from left to right on the coordinate plane.

How It Works

When you have the graph of a function, place a horizontal line across the graph. Sometimes it is easiest to use a thin ruler or piece of paper as the line and hold it up against the graph. Be sure to move it up and down along the y-axis to hit all points of the graph. It is possible that the horizontal line will touch the graph in one place in some areas but in two places in other areas.

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If the horizontal line only crosses the graph of the function in one place, then the function is one-to-one. This is because there is only one x-value for each y-value.

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If the horizontal line crosses the graph in more than one place, it is not a one-to-one function, because there is more than one x-value for certain y-values.

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