In this explainer, we will learn how to identify the domain of a rational function and the common domain of two or more rational functions.
Let us recall what we mean by the domain of a function. When we define a function, we typically write it in the form . What this means is that for every , the function maps it to a . We write this as . Here, we call the domain of the function and the codomain of the function. Let us illustrate this idea in the diagram below.
Here, the domain of is , and the codomain is . Additionally, the range of , which we denote by , is , which in this case is smaller than the codomain. If we wanted to write that 1 is mapped to 2 by , we would write that mathematically as .
To take an example that uses a specific equation, suppose we have the function defined by
Here, we can safely define the domain of this function to be the set of real numbers , since we can take any that is a real number and put it into the equation with no problems. We can also define the codomain of this function to just be , since for any , we know that . We do not need to know it here, but the range of this function is . We know this because for all , so the value of cannot go below 7. On the other hand, as gets bigger, will continue to increase, eventually going to .
Recall that a polynomial function takes the form
The above expression is a second-order polynomial, or a quadratic. Typically for any polynomial function, we want both . However, as we will see later on, for different functions, we may sometimes need to restrict the domain, depending on what our function is.
Recall that a rational number (or a fraction) is any number of the form , where and are integers and . It is important to remember that the denominator cannot be zero since dividing by zero is an undefined operation. We can extend this idea of rational numbers to rational functions, with the following definition.
Definition: Rational Functions
A function is called a rational function if it can be written in the form where and are polynomial functions and for all .
We note that the domain of has to exclude any points where . Since is a polynomial, this sometimes means we have to find its zeros to find out which points are not valid. For example, let us say we had the function
We can say that this function is not valid if , which we can easily solve to find that cannot be equal to 2. Consequently, the domain of this function must be , that is, the set of all real numbers excluding 2. We note that the codomain of this function is still , and generally speaking any rational function will still have codomain .
Let us consider a more complicated example where we have to solve a quadratic equation to find which values are not defined.
Example 1: Determining Which Values Are Undefined for a Rational Function
For which values of is the function not defined?
To find for which values is not defined, we only have to consider the denominator of the fraction, , and when it is equal to 0. This is because whatever values the numerator takes, it will not cause to be undefined.
So let us attempt to solve . To solve a quadratic equation, we can either attempt to factor it or use the quadratic formula. Let us attempt to factor it here by supposing
We can see that and have to add up to get 12 and multiply to get 32. Trying different numbers, we can see that taking and works, or vice versa. Consequently, we have
Thus, or are the solutions to the equation, and we can say that the original function is not defined for .
Let us attempt another similar problem, this time in an example where we have to find the domain of the function.
Example 2: Determining the Domain of a Rational Function
What is the domain of the function ?
To work out the domain of this function, we need to find out the values for which it is not defined, which means we have to consider the points where the denominator .
To solve a quadratic, we usually want to factor it, or apply the quadratic formula. In this example, however, this may not be possible. Indeed, if we rearrange the above equation, we see that
Since is negative, we cannot take the square root of it (provided ). This means that this equation has no solutions. In other words, there is no point at which , and so the rational function is defined for all values. Thus, the domain of is .
We have now looked at some examples where we have looked at rational functions and calculated the points where they are not defined (and by extension their domains). We might also encounter problems where we are given the domain of a function and have to find what values fit the domain. Let us consider an example that uses this concept.
Example 3: Finding the Value of a Function given the Domain
Given that the domain of the function is , evaluate .
This question has a couple of parts to it. We must first try to find the points at which cannot be evaluated, which should allow us to find the unknown variable . Then, we can substitute into to find the answer.
To start with, we can see that is the sum of two separate rational functions. To calculate the points at which cannot be evaluated, and hence its domain, we simply have to consider the points at which either function cannot be evaluated on its own, since in that case the sum will not be valid either. Let us do this first with
This is a simple rational function, and we can see that when the denominator , it is not valid. Next, we have
Here, we see that the function is not valid when the denominator . Rearranging gives us .
Now, the domain of the function is , which means cannot be evaluated at or . We can see that the requirement comes from . Thus, must lead to . Combining these equations gives us or . This gives us
To solve the problem, we finally want to evaluate . Having worked out what is, we can simply substitute 3 into the equation:
In the last example, we touched on the idea that combining multiple rational functions together means that the domain of the resulting function has to take the domain of each separate function into account. Tangential to this is the idea of considering the common domain of two or more rational functions. As a simple example, suppose we had the following functions:
To consider the common domain here, we simply take the intersection of the two domains. That is to say, the common domain of these two functions is
In general terms, we can take any number of functions, and the common domain is simply the intersection of those domains. Therefore, all we have to do is work out the domain of each function individually and combine them all together. Let us look at an example that covers this concept.
Example 4: Determining the Common Domain of Three Rational Functions
Find the common domain between the functions , , and .
To find the common domain of these functions, we have to work out the domain of each separately and then take their intersection. First, let us consider
Considering the denominator , we can see that is not defined when , since we would be dividing by 0. Therefore, the domain of this function is everywhere excluding this point, which is . Next, we have
In a similar manner, we see that is not defined when , so the domain must be . Finally, we consider
is not defined when the denominator is equal to 0. This function is a cubic polynomial; however, because it has a common factor of , we can factor it fairly easily as follows:
Setting this equal to 0, we see that is not defined for , or . So, the domain is .
Finally, we must combine the domains together by taking their intersection. This intersection is simply minus all of the points we took out from each function. In summary, the common domain is
Finally, let us look at one more example that combines some of the concepts we have seen so far.
Example 5: Finding the Value of a Function given the Common Domain
If the common domain of the two functions and is , find the value of .
The common domain is the intersection of the domains of and . In order to determine what is, we will begin by calculating the domains of and and comparing these values to .
For starters, let us consider
To find the domain of a rational function, we want to find when the denominator is equal to 0. In this case, we see that
As must be real valued, we conclude that this equation has no solutions; that is, for all . Thus, the domain of is simply .
Now, before calculating the domain of , let us note that since we know the common domain is , we have
Since every point in the domain of belongs to , we have
This shows us that both singularities and must come from . Now, is
As with , the domain of this function can be found by considering the points where the denominator of is equal to 0. In other words, we need to solve
Normally, we would only be able to solve this quadratic in terms of . However, since we know that the domain of , the solutions of this equation must correspond to the singularities and .
A quadratic with solutions and must be of the form
Comparing this to the denominator of , we have
Thus, we find .
Let us finish by going over the key points we have learned in this explainer.
- A function is called a rational function if it can be written in the form where and are polynomial functions and for all .
- We can identify the domain of a rational function by solving in the denominator and excluding those points from .
- The common domain of two or more rational functions can be found by taking the intersection of their domains.