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. null null 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. null 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.
