Math Lesson 16.2.6 - Horizontal Line Test

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Welcome to our Math lesson on Horizontal Line Test, this is the sixth lesson of our suite of math lessons covering the topic of Injective, Surjective and Bijective Functions. Graphs of Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Horizontal Line Test

The horizontal line test is a method used to check whether a function is injective (one-to-one) or not when the graph of the function is given. It consists of drawing a horizontal line in doubtful places in order to 'catch' any double intercept of the line with the graph. In that case, there is a single y-value for two different x-values - a thing which makes the given function unqualifiable for being injective and therefore, bijective. Therefore, such a function can be only surjective but not injective. Let's see an example to clarify this point.

Example 6

Use the horizontal line test to determine which of the following graphs is/are injective.

Solution 6

1. This is an injective function because the only part of the graph in which there may be some doubt about the intersection in more than one point with a horizontal line is at y = 0, which corresponds to the horizontal axis. However, looking at the graph carefully, we can see that there is a single x-intercept of the graph (at x = 1). Therefore, this function is injective. Indeed, if we can write this function in the factorised form as f(x) = (x - 1)³ using the transformation (a - b)³ = a³ - 3a²b + 3ab² - b³, it becomes clearer that this polynomial function has a single root, x = 1. This means the horizontal axis can't intercept the graph at more than one point.
   As for the rest of the graph, there is no doubt that a horizontal line drawn from anywhere intercepts the graph at a single point. Therefore, this function is injective (one-to-one), i.e. every x-value has in correspondence a single y-value.
2. This is not an injective function because it is evident that all lines drawn from any vertical position intercept the graph in two points. For example, the horizontal axis (y = 0) intercepts the graph at x = -1 and x = 1 because
   f(-1) = 1/(-1) - (-1)
   = -1 + 1
   = 0
   and
   f(1) = 1/1 - 1
   = 1 - 1
   = 0
   Look at the figure.
3. This too is not an injective function because we can find more than two x-values for the same y-value. Graphically, this is discovered by making the horizontal line test. Again, we can choose the X-axis but since we have used it in the previous two examples, we can choose another horizontal line, for example, y = 1. After doing this, it is easy to see that this line intercepts the graph at three different points, as shown in the figure.

You have reached the end of Math lesson 16.2.6 Horizontal Line Test. There are 7 lessons in this physics tutorial covering Injective, Surjective and Bijective Functions. Graphs of Functions, you can access all the lessons from this tutorial below.

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