Explainer: Horizontal and Vertical Asymptotes of a Function

In this explainer, we will learn how to find the horizontal and vertical asymptotes of a function.

Before we look explicitly at how to find an asymptote of a rational function, let us recall what an asymptote is.

Definition: Asymptote

An asymptote is a line that approaches a given curve arbitrarily closely. This is illustrated by the graph of 𝑦=1π‘₯.

Here, the asymptotes are the lines π‘₯=0 and 𝑦=0.

In order to identify vertical asymptotes of a function, we need to identify any input that does not have a defined output, and, likewise, horizontal asymptotes can often be identified by considering outputs that cannot be reached from any input in the function's domain. In particular, for rational functions, knowing the domain and range will help us to identify the asymptote and vice versa. Therefore, if we are looking to identify the asymptotes of a rational function, we take a very similar approach to how we identify the domain and range of a rational function.

If we consider the function 𝑦=33βˆ’π‘₯,

we need to find if it has any undefined input and, equally, any values that do not exist in its range. We know that a rational function is undefined when its denominator is zero. Here, 3βˆ’π‘₯ is zero when π‘₯=3.

This, therefore, is the equation of an asymptote. Are there any others? Well, if we consider all the possible outputs of the function and consider what happens as the inputs get progressively large, we can see that the outputs of the function get closer and closer to zero but can never actually get there. Therefore, we have another asymptote at 𝑦=0.

By identifying the asymptotes of a rational function, we can easily identify the domain and range. The function has a domain, which is all of the real numbers except 3, and a range, which is all of the real numbers except 0.

Let us now look at a couple of examples.

Example 1: Identifying the Asymptotes of Rational Functions

Determine the vertical and horizontal asymptotes of the function 𝑓(π‘₯)=βˆ’1+3π‘₯βˆ’4π‘₯.

Answer

To find the vertical asymptotes of the function, we need to determine if there is any input that results in an undefined output. The function 𝑓(π‘₯) contains two rational expressions that are undefined when their denominators are zero. The expression 3π‘₯ is undefined when π‘₯=0, and the expression βˆ’4π‘₯, again, is undefined when π‘₯=0. Therefore, we have a vertical asymptote with the equation π‘₯=0.

To find the horizontal asymptotes, we need to find if there are any values that do not exist in the range of the function. If we look at the outputs of the function as the inputs get progressively large, we can see that the expressions 3π‘₯ and βˆ’4π‘₯ get closer and closer to zero and the function gets closer and closer to βˆ’1 but never actually achieve this value. Thus, we have a horizontal asymptote with the equation 𝑦=βˆ’1.

Before we look at the next example, it is worth noting that a hyperbola is a type of rational function with two asymptotes.

Example 2: Identifying the Asymptotes of Rational Functions

What are the two asymptotes of the hyperbola 𝑦=84π‘₯βˆ’3+53?

Answer

To find the vertical asymptotes of the function, we need to determine if there is any input that results in an undefined output. The function contains a rational expression that is undefined when its denominator is zero. The expression is undefined when 4π‘₯βˆ’3=0 and, therefore, has an asymptote with the equation π‘₯=34.

To find the horizontal asymptotes, we need to find if there are any values that do not exist in the range of the function. If we look at the outputs of the function as the inputs get progressively large, we can see that the expression 84π‘₯βˆ’3 gets closer and closer to zero and the value of the function gets closer and closer to 53 but will never actually achieve this value. Thus, we have a horizontal asymptote with equation 𝑦=53.

As previously mentioned, if we identify the asymptotes of a rational function, we can use this information to easily find the domain and range of the function, and, equally, we can use the information about the asymptotes to help us sketch or identify the graph of the function. Let us now look at examples of this.

Example 3: Using the Asymptotes of a Rational Function to Find Its Domain and Range

Determine the domain and the range of the function 𝑓(π‘₯)=1π‘₯βˆ’5 in ℝ.

Answer

In this question, we are fortunate to have been given the graph of the rational function. This allows us to easily identify the equations of the asymptotes: we can see that the equation of the vertical asymptote is π‘₯=0, and that the equation of the horizontal asymptote is 𝑦=βˆ’5. Using this information, we can state that the domain of the function is β„βˆ’{0} and that the range of the function is β„βˆ’{βˆ’5}.

Suppose we were not given the graph of the function. We could identify the asymptotes by looking for any input that results in an undefined output and looking for any output that cannot be achieved regardless of the input. Here, we can see that we have an undefined output for an input of zero as 10 is undefined. Hence, we have an asymptote with the equation π‘₯=0. We can also see that as π‘₯ gets progressively large the outputs of the function get closer and closer to βˆ’5 but can never take this value. Therefore, we also have an asymptote with the equation 𝑦=βˆ’5.

Example 4: Finding the Asymptotes of a Function in order to Identify Its Graph

Which of the following graphs represents 𝑓(π‘₯)=1π‘₯+1?

Answer

In this question, we can identify the graph of the rational function by determining the position of its asymptotes. If we look at the equation of the function, we can identify any vertical asymptote by identifying any input that results in an undefined output. The function is undefined when the denominator is equal to zero, that is, when π‘₯+1=0. Therefore, we have an asymptote with the equation π‘₯=βˆ’1.

To find the horizontal asymptotes, we need to find if there are any values that do not exist in the range of the function. As the inputs get progressively large, we can see that the function 1π‘₯+1 gets closer and closer to zero, but it will never actually achieve this value. Thus, we have a horizontal asymptote with the equation 𝑦=0.

Using this information, we can see that the correct graph is (c).

Example 5: Identifying the Domain of a Rational Function

Find the domain of the function 𝑓(π‘₯)=π‘₯βˆ’36π‘₯π‘₯+6π‘₯.

Answer

This is a particularly interesting question as it is not immediately obvious how the graph matches up with the function. It is very easy with a question like this to mistakenly state the domain and range and equally make assumptions about the nature of the function, including whether it has any asymptotes.

If we factor the numerator and denominator of the function, we get that 𝑓(π‘₯)=π‘₯(π‘₯+6)(π‘₯βˆ’6)π‘₯(π‘₯+6).

At this point, we can see that the function is not defined at two points: π‘₯=0 and π‘₯=βˆ’6. If you input either of these values, you will get 00, which is not defined. Using what we know about rational functions, it would be fair to assume that the function, therefore, has asymptotes at these two points. This, however, is not the case. Providing π‘₯ does not equal 0 or βˆ’6, then we can simplify our function as follows: 𝑓(π‘₯)=π‘₯(π‘₯+6)(π‘₯βˆ’6)π‘₯(π‘₯+6),

which simplifies to 𝑓(π‘₯)=π‘₯βˆ’6.

This shows us that for all values of π‘₯, excluding 0 and βˆ’6, the function simplifies to a line. At this point, we can determine that the domain of the function is β„βˆ’{βˆ’6,0}.

The common mistake here is to simplify the function first and then state that the domain is the whole real numbers, but this is not the case.

Notice that in this example, the domain of the function does not tell us about the asymptotes of the function. In fact, the function has no vertical or horizontal asymptotes. This will be the case where we have rational functions defined by an expression that can be simplified by canceling common factors in the numerator and denominator.

Key Points

  1. To find the vertical asymptotes of the function, we need to identify any point that would lead to a denominator of zero, but be careful if the function simplifiesβ€”as with the final example.
  2. To find the horizontal asymptotes of a rational function, we need to identify any value that the function cannot take. It can often be helpful to look at the limits of the function to aid you in this process.
  3. We can use the asymptotes to help us identify the range and domain of the function.
  4. We can use the asymptotes to help us sketch or identify the graph of the function.

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