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General Studies Math: Help & Review8 chapters | 85 lessons

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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.

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 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.

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.

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.

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.

Why is it important to know if a function is one-to-one? When dealing with mathematical functions, it is helpful to understand all of the properties of that function. It can give you insight into the best ways to solve problems dealing with that function, especially if they are higher-level problems that involve many different functions working together. Injective functions are also important when dealing with situations that might need to be reversed.

For example, if you have traveled a path into the forest, it is very important that you can retrace your path back out of the woods. It wouldn't be very helpful to have traveled a path only to find that the path back took you to a completely different place. Injective functions also are important in computer programming. There are some programs that you want to work the same each and every time. 'When you type the letter 'a' on your computer keyboard, you want an 'a' to show up every time, without fail. You don't want to be surprised at the letter that appears. You want that one-to-one relationship.

Let's review! The **horizontal line test** is a simple mathematical tool that is used to determine if a function is a **one-to-one function (injective function)** or not. The test is performed by holding a horizontal line up against the graph of a function. If the horizontal line only touches the graph at one point all along the graph, then it is a one-to-one function

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General Studies Math: Help & Review8 chapters | 85 lessons

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