Explainer: Domain and Range of a Rational Function

In this explainer, we will learn how to find the domain and range of a rational function either from its graph or its defining rule.

Before we start looking at how to find the domain and range of rational functions, let us remind ourselves what we mean when we talk about the domain and range of a function.

If we think of a function as a mapping that takes an input to an output, the domain would be the set of inputs and the range the set of outputs. Consider the following mapping diagram:

We can see the inputs on the left and the outputs on the right. Here, the domain is the set {1, 2, 3, 4} and the range is the set {2, 4, 6, 8}. If we consider the function 𝑓(π‘₯)=3π‘₯+2 with domain {3, 5, 7, 9}, then we can calculate the range by substituting each of the values from the domain into the function: 3(3)+2=11,3(5)+2=17,3(7)+2=23,3(9)+2=29.

The range is, therefore, the set {11, 17, 23, 29}.

Before moving on, let us recall that

  • β„• is the set of natural numbers.
  • β„€ is the set of integers.
  • β„š is the set of rational numbers.
  • ℝ is the set of real numbers.
  • β„‚ is the set of complex numbers.

If we consider the function 𝑓(π‘₯)=4π‘₯βˆ’2 with domain π‘₯βˆˆβ„ (which means π‘₯ belongs to the set of real numbers), it can be helpful when thinking about the range of the function to consider its graph.

Here, we can see the graph is a straight line, and every real number input has a real number output, and since the line continues infinitely in both directions, any real number output is possible. Therefore, the range is all the real numbers.

If we look at a quadratic function, for example, 𝑓(π‘₯)=π‘₯ whose domain is the real numbers, and if we look at the graph of 𝑦=𝑓(π‘₯), we can use this to determine the range.

We can see that for any input, the output is positive, and therefore the range of the function is any real number greater than or equal to zero.

Now, given this recap, let us introduce the concept of rational functions. Generally, we tend to define the domain and range of functions over the real numbers and we will do likewise here. We will take a different approach to working out the domains and ranges of rational functions, as it is not always easy to sketch their graphs. Consider the function 𝑓(π‘₯)=2π‘₯+3.

Notice, for an input of βˆ’3 we get 20.

Any division by zero is undefined, so we have that the function is undefined at this point. Any nonzero input, however, will have a corresponding output in the real numbers, so we can state that the domain is the real numbers excluding βˆ’3, written β„βˆ’{βˆ’3}.

By considering the nature of the function, we can also see that any real number output can be achieved with the exception of zero: as π‘₯ gets progressively larger in magnitude, the output gets progressively smaller; however, the output can never actually reach zero. Therefore, the range of the function is β„βˆ’{0}.

In general, to calculate the domain of a rational function, we need to identify any point where the function is not defined, that is, any point that would give a denominator that is equal to zero. To find the range of a rational function, we need to identify any point that cannot be achieved from any input; these can generally be found by considering the limits of the function as the magnitude of the inputs get very large. Let us look at some examples.

Example 1: Finding the Domain and Range of a Rational Function with One Unknown in the Denominator

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

Answer

From looking at the graph, it looks like the domain is β„βˆ’{5} and that the range is β„βˆ’{0}. However, we should also check this algebraically.

We know that a rational function is not defined for any input that results in a zero denominator. We can equate the denominator of the function to zero to find the undefined point. We have that π‘₯βˆ’5=0, which gives a solution of π‘₯=5.

This confirms that the domain is β„βˆ’{5}. To confirm the range we need to identify all values that cannot be achieved given the domain. As π‘₯ gets progressively larger in magnitude, the output tends to zero but will never actually reach zero. Therefore, the range is indeed β„βˆ’{0}.

Let us now look at an example where we are not given the graph and have to approach the question algebraically.

Example 2: Finding the Domain and Range of a Rational Function Algebraically with One Unknown in the Denominator

Determine the domain and range of the function 𝑓(π‘₯)=1π‘₯βˆ’2.

Answer

Remember that the expression 10 is not defined, and from this we can determine that a rational function is not defined for any input that results in a zero denominator. We can equate the denominator of the function to zero to find the undefined point. We have that π‘₯βˆ’2=0, which gives a solution of π‘₯=2.

Therefore, we can state the domain as β„βˆ’{2}. To find the range, we need to identify all values that cannot be achieved given the domain. As π‘₯ gets progressively larger in magnitude, the output gets progressively close to zero but will never actually reach zero. Therefore, the range is β„βˆ’{0}.

Now, let us look at an example of finding the domain and range of a function with an unknown on the top and the bottom of the expression.

Example 3: Finding the Domain and Range of a Rational Function Algebraically with an Unknown in the Numerator and Denominator

Define a function on the real numbers by 𝑓(π‘₯)=2π‘₯+34π‘₯+5.

  1. What is the domain of the function?
  2. Find the one value that 𝑓(π‘₯) cannot take.
  3. What is the range of the function?

Answer

Part 1

To find the domain of the function, we need to establish if there are values of π‘₯ for which 𝑓(π‘₯) is undefined. As this is a rational function, it will be undefined when its denominator takes a value of zero. Therefore, the graph of the function would have an asymptote when 4π‘₯+5=0. If we subtract 5 from each side of the equation and then divide through by 4, we find that the asymptote has the equation π‘₯=βˆ’54. Therefore, the domain of the function is all real numbers except βˆ’54, notated β„βˆ’ο¬βˆ’54.

Part 2

In order to determine the value that 𝑓(π‘₯) cannot take we need to explore the limit of the function. That is, what happens as π‘₯ gets large. To make this process easier it is helpful to divide the top and bottom of the function by π‘₯ to get 𝑓(π‘₯)=2+4+.οŠ©ο—οŠ«ο—

From here, we can see that as π‘₯ gets progressively large, 3π‘₯ and 5π‘₯ get closer and closer to zero, and hence the function gets closer and closer to 12 but never actually reach it.

Part 3

From the solution to part 2, we can see that the range of the function is all real numbers except 12, notated β„βˆ’ο¬12.

Let us now look at a couple of more complicated examples. Firstly, a question where the rational function is presented as a sum of two functions, and, secondly, a rational function whose numerator and dominator are nonlinear.

Example 4: Finding the Domain of a Sum of Rational Expressions

Determine the domain of the function 𝑓(π‘₯)=3π‘₯βˆ’3+1π‘₯+4.

Answer

Recall that rational functions are defined when their denominators are nonzero. From the function written in this form, we can see that there are two points at which the function is undefined: when π‘₯βˆ’3=0 and when π‘₯+4=0. This means that the function is undefined when π‘₯=βˆ’4 and π‘₯=3. Therefore, the function’s domain is all the real number except βˆ’4 and 3, notated β„βˆ’{βˆ’4,3}.

As an additional piece of information, if we were trying to find the range of this function, even though each of the rational expressions being summed cannot take a value of zero themselves, there does exist an input π‘₯ that is mapped to zero, which is π‘₯=βˆ’94. Therefore, the range of this function is actually the whole real numbers (ℝ).

Example 5: Finding the Domain of a More Complicated Rational Expression

Find the domain of the real function 𝑓(π‘₯)=π‘₯βˆ’1610π‘₯+70π‘₯.

Answer

Remember that a rational function is only defined when its denominator is nonzero. Therefore, to find the domain, we need to find the zeros of the equation 10π‘₯+70π‘₯=0. To solve this, we can factor out an π‘₯ to get π‘₯ο€Ή10π‘₯+70=0.

From here, we can see that we have a zero when π‘₯=0. The quadratic 10π‘₯+70, however, has no real roots. Therefore, the only zero of the denominator is π‘₯=0. The domain is, thus, all real numbers except 0, notated β„βˆ’{0}.

Key Points

To find the domain and range of rational functions remember the following steps:

  1. To find the domain of a rational function, we need to identify any points that would lead to a denominator of zero.
  2. To find the range of a rational function, we need to identify all values that the function cannot take. It can often be helpful to look at the limits of the function to aid us in this process.
  3. It can be helpful to consider the graph of the function to aid in the process of identifying the domain and range of a function.

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