In this explainer, we will learn how to multiply and divide rational functions.
Let us recall the definition of a rational function.
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 .
Recall that the domain of a rational function depends upon the denominator. If , then cannot take values such that because then we would be dividing by zero and would be undefined.
Let us consider what happens when we multiply two rational functions together. Recall that if we had two ordinary rational numbers (or fractions) given by and , their product is simply
That is to say we multiply the numerators (i.e., the tops) together and the denominators (i.e., the bottoms) together. Rational functions work in exactly the same way. Suppose we have two rational functions and . Then, their product is
The only difference is that we are now dealing with functions that depend on rather than just numbers. Naturally, we still need to consider what domain is appropriate for this function. Using what we know about rational functions, we can say that is only valid when the denominator , which means we need both and (since if either of them were zero, their product would be zero too). This leads to the following rule.
Rule: Product of Rational Functions
Let and be two rational functions, and suppose their product is . Then, and the domain of is the common domain of and .
Recall that the common domain of two rational functions is just the intersection of their domains, and we can calculate this by working out all the points that cause either of the denominators or to be zero and subtracting those points from .
Let us demonstrate how this works with a simple example. Suppose we have
To work out their product, , we simply multiply the numerators and the denominators together.
To work out the domain of this function, we find the common domain of and . The domain of is , and the domain of is . Therefore, the domain of is the intersection of these two domains, .
We can verify this by looking at the denominator of . We can see that if we took or , the denominator would be 0, meaning the expression would be undefined. Otherwise, any other value for would be valid.
It is important to note that we always need to check that the domain is valid before simplifying by canceling any terms. Suppose, for example, we had
If we simplify this function before checking the domain, we have
At first, it looks like this function is valid for any . But this is discounting the fact that is undefined if we take or in the original expression. So, it is crucial that we check for any points where the denominator is 0 before canceling any terms.
Now, up to this point, we have only considered rational functions containing linear polynomials, but keep in mind that we will also need to consider rational functions that involve higher-order polynomials like quadratics as well. In such a situation, the best approach is usually to simplify the expressions via factoring as much as possible before multiplying them together.
Recall that, to factor a quadratic, we can apply the following general process.
How To: Factoring a Quadratic Expression
Let be a quadratic function. Then, we can apply a step-by-step process to factor as follows:
- Find and what factors it has. So, look for numbers and such that .
- Find two factors of that add together to get . Let us say we find that .
- Rewrite the expression as .
- Factor the first two terms and the last two terms.
- Finally, factor out the common factor.
So, let us apply this to an example and say we have . Here, , , and . Applying the above process, we have
- . Its factors are , , and .
- Adding these factors together, Since , this means 3 and 4 are values that work.
- We rewrite the expression as .
- We factor out from the first two terms and 2 from the second two terms to get .
- We factor out as a common factor to get .
Finally, we can verify that this factoring is correct by multiplying out the parentheses, which shows that this is indeed equal to . In addition to this method, we also note that sometimes, we can apply shortcuts, such as noting , which saves us a bit of working.
So, let us consider a product of rational expressions involving quadratic terms and see the general process that we use.
Example 1: Simplifying a Rational Function Involving the Product of Two Rational Expressions and Finding the Domain of the Resulting Function
Simplify the function , and determine its domain.
The best way to approach a question like this is to begin by simplifying the expression by factoring it where possible. Looking at the components of this expression one-by-one, we note that the numerator of the first rational function is a perfect square quadratic and can thus be factored by
In the denominator, we can factor out to find that
In the second rational function, we can factor out 7 to get
Finally, in the second denominator, we have a quadratic that is the difference of two squares and can be factored by
Having factored all the terms, we can rewrite as follows:
Before we multiply these expressions together, it is important that we note that, for the first expression, the points and are invalid, and for the second, the points and are invalid. Using the fact that the domain of the product of rational functions is the common domain of those functions, we therefore know that the domain of must be .
We note that it is important to do this before simplifying further, because even if we cancel out terms, we are still not allowed to let have values that are invalid in the original expression.
Now, before multiplying the two expressions together, we note that we can already cancel out terms in the numerators and the denominators. Noting that in the denominator and that we can take the term outside the fraction, we have
Now, we multiply the fractions together as follows:
So, in conclusion, we find that and its domain is .
Let us consider a similar example, this time with a cubic expression in the numerator, and apply the same concepts we have just used.
Example 2: Simplifying a Rational Function Involving the Product of Two Rational Expressions and Finding the Domain of the Resulting Function
Simplify the function , and determine its domain.
Let us start by factoring the two expressions as much as possible. Going through the numerators and denominators one by one, we start with
This is a cubic expression. Recall that, for factoring the sum of cubes, we have the following formula we can use:
Let us determine if our function is of the correct form (i.e., is there a such that ?). Going through , we see that . This means that and we can apply the formula to obtain
We note that, as is a sum of cubes, it only has one real solution, . Hence, the other factor of the expression, , has no real solutions and cannot be factored further.
Let us consider the other terms in . The denominator of the first term can be factored by taking out :
In the numerator of the second term, we have , which is already in its simplest form, and in the denominator, we once again have , which we know has no real solutions. All in all, this gives us
Before simplifying further, let us find what values is valid for in the domain. In the first fraction, we have , which means and . For the second, we know that has no real solutions, so any value for is fine. Therefore, the domain of is .
Let us now multiply and simplify where possible:
So, in the end, we find that , with domain .
While we often have to simplify and find the domain of the product of rational expressions, sometimes we are just asked to find the value of that product at a certain point. In this situation, we simply have to substitute the given value into the function. It is important to pay attention to what the question is asking so that we can avoid doing work that is actually not necessary.
Example 3: Simplifying a Rational Function Involving the Product of Two Rational Expressions and Evaluating the Resulting Function at a Given Value
Given the function , evaluate , if possible.
Normally, for questions where we have to multiply two rational functions together, it is easiest to factor the equations first so that we can find their domains and cancel any common factors. In this situation, though, we only need to evaluate at a single point, so we can just substitute into and see what we get. For completeness, however, we will demonstrate what happens when you factor the equation. After some calculations, we will get
We note here that, due to the factors in the denominators, the domain of is . Since is not in the domain, this shows that evaluating would result in an undefined quantity. If we were to continue further and cancel out the factors, we would get
Now, if we had not checked the domain earlier and tried to evaluate at , we would have found
Even though we get a well-defined answer here, we are not actually evaluating at 7 but rather a modified version of with the singularity cut out. So, the answer is not valid. As discussed earlier, the easiest approach to this type of question is simply to substitute into the original equation, which gives us
As we already knew, the result is undefined since it is , meaning we cannot evaluate the function. Therefore, is undefined.
A natural extension to the topic of multiplying rational functions is the multiplicative inverse of a rational function. Recall that for any real number, its multiplicative inverse is a number that multiplies that number to make 1:
Here, is the multiplicative inverse of . For real numbers, we can find this number by taking the reciprocal, which is just 1 divided by the number.
Additionally, we can consider the multiplicative inverse of a rational number:
In this case, the reciprocal of is , which is . In other words, the multiplicative inverse can be found by swapping the numerator and the denominator.
When it comes to the multiplicative inverse of a rational function, the idea is exactly the same. Suppose we have the rational function . Then, we have
That is to say, the multiplicative inverse of is , which can be found by swapping the numerator and the denominator.
The only complication we need to consider is the domain of the multiplicative inverse. If we consider the function , we can see that cannot be equal to zero or the denominator would be zero, leading to the function being undefined. However, in order to find the multiplicative inverse in the first place, we need to take the reciprocal; if we consider the expression , we can see that if , then will be undefined, meaning the inverse cannot be defined either. This leads to two requirements in the domain of the multiplicative inverse, which we summarize in the following rule.
Rule: Multiplicative Inverses of Rational Functions
Recall that the zeroes of a function are the points , where . If is a rational function, then its multiplicative inverse is and its domain is , where denotes the set of zeroes of a function.
This rule means we need to consider both the domain of the original function and that of the new function, by checking any points where they can be equal to zero.
To properly consider this complication with the domain, let us explore an example that deals with this issue.
Example 4: Finding an Unknown Value given the Domain of the Multiplicative Inverse of a Rational Function
Given that the domain in which the function has a multiplicative inverse is , find the value of .
To answer this question, we will need to recall how the domain of the multiplicative inverse of a rational function is calculated. First of all, the multiplicative inverse can be found by taking the reciprocal:
First, we can see that we will need ; otherwise, the fraction will not be defined. Additionally, must also be defined (otherwise, will not be defined), and by considering the denominator of , we find that . This means that we have two requirements in the domain:
This corresponds to the points where either the numerator or the denominator of the original function can be zero (i.e., the zeroes). Therefore, the domain is .
If we compare this to the given domain, we can see that the two sets are equivalent when . Thus, .
Now that we have seen how the calculation of the domain works, let us explore the full process of finding the multiplicative inverse of a rational function.
Example 5: Finding the Multiplicative Inverse of a Rational Function and Its Domain
Given that , find its multiplicative inverse in its simplest form, showing its domain.
The best way to approach this type of problem is to begin by factoring the rational function, since that will allow us to simplify easily and identify zeroes in the denominator. Considering the numerator, we can see that it is a difference of squares, so it can be factored as follows:
For the denominator, since both terms share a common factor of , we can factor to get
Therefore, the factored form is
Before simplifying further or finding the multiplicative inverse, we need to find the zeroes of the denominator. These are the values of that result in , which are and .
Now, we can simplify:
To find the multiplicative inverse, we take the reciprocal, which means swapping the numerator and the denominator, giving us
For this function, we can see that leads to the denominator being zero. The domain of the function is therefore minus this value, along with the values that cause to be undefined that we calculated previously, that is, .
In summary, the multiplicative inverse is , with domain .
Having just considered multiplicative inverses, let us consider a very similar concept: division. When we divide by a number, we are actually multiplying by the multiplicative inverse:
This can of course be extended to rational numbers. Recall that if we want to divide by a rational number, we can in fact multiply by the reciprocal (i.e., the multiplicative inverse):
We can do the same thing with rational fractions.
Supposing we have two rational functions and , their quotient is
Once we take the reciprocal of the function, the rest of the calculation goes the same way as for multiplying functions together.
However, much like for multiplicative inverses, we do need to be careful when considering the domain of the quotient. In the above expression, we first of all need and for any . Once we have taken the reciprocal, we can see that we also need . So, in total, when compared to the previous cases, we need to check an extra term in order to determine the correct domain. This leads to the following rule.
Rule: Quotient of Rational Functions
If and are two rational functions and their quotient is , then and the domain of is , where denotes the set of zeros of a function.
This rule means we need to consider all the points where a denominator might be 0 and exclude them from the domain.
We will demonstrate exactly how this works in the following example.
Example 6: Identifying the Domain of the Quotient of Two Rational Expressions
Determine the domain of the function .
Recall that, for the domain of the quotient of two rational expressions, we need to calculate the points where
We do not need to consider the equation since it will not cause problems when dividing. By looking at the given function, we can confirm that if or , will be undefined. This means that we cannot have , since this leads to both expressions being 0.
We know that when we divide by a rational expression, that is the same as multiplying by the reciprocal. So, another way to write is
We note that, in order to find the domain, it is important not to simplify the expression further since we might cancel terms that are important. In this example, we have in the denominator and in the numerator, leading to canceling out. If we performed this before checking the domain, then we might miss the fact that (although, fortunately, we have already found this above).
In any case, by considering this form for , we can confirm that is the other requirement for our domain, which leads to . Now knowing that the set of invalid points for this function is , we can say that the domain must be .
Having looked at the division of rational functions with linear components, let us once again return to the case of a problem involving quadratics, since this will result in us having to do slightly more in-depth calculations.
Example 7: Evaluating the Domain of the Quotient of Two Rational Expressions with Quadratic Components
Determine the domain of the function .
Recall that, for the domain of the quotient of two rational expressions, we need to consider the points where
We do not need to consider since it is not a problem if it is equal to 0. So, let us begin by factoring these three expressions. We can factor the first expression to get
So, here we find are two points that are not valid in the domain. For our second expression,
So, is another invalid point. Finally, we have
This tells us is invalid, which we already knew from the first expression.
Combining this together, we find that the domain of is .
As of yet, we have not had to calculate an explicit expression for the quotient of two rational functions. For our final example, let us consider a problem where we must do this and then find the point at which the resulting function is equal to a given value.
Example 8: Simplifying a Rational Function Using Factorization and Then Evaluating Its Variable at Given Values
Given that and , find the value of .
To solve this problem, we could try substituting into directly and setting it equal to 4, but as this would result in an equation in terms of that would not simplify easily, this would probably prove difficult. Instead, a better approach is to begin by simplifying the function and then substituting into the simplified form. Let us begin the simplification by starting with the top left numerator,
To factor this, since it is a quadratic expression, we want to find two numbers that multiply together to get 14 and add together to get 9. These are 2 and 7. Thus, we can factor the equation to get
For the denominator, we notice that is a difference of two squares. Thus, we can factor it to find
For the second fraction, we can see that, in the numerator, we again have a difference of two squares that can be factored to get
Finally, for the denominator, we notice there is a common factor of , which we can factor out to get
Putting this all together, we get the following factoring of :
Now, we have been given that is a valid equation, but just to be safe, we can find the domain of before simplifying further, by calculating the zeros of the following equations:
The first equation gives us and , the second gives us and , and the third gives and again. Thus, the domain of is .
Now, we simplify the equation for by taking the reciprocal of the divisor to get
Next, we cancel out the factors as follows:
Thanks to the simplifications, this equation has now been significantly reduced to the point where we can solve quite easily. We have
Let us finish by going over the key points we have learned during this explainer.
- Multiplying rational functions together works as follows: where the domain of the product is and denotes the zeros of a function.
- To find the multiplicative inverse of a rational function , we swap the numerator and the denominator: and the domain of the inverse is .
- We can divide rational functions by multiplying by the inverse: where the domain of the quotient is .
- Factoring helps us to simplify the answer and find any points in the domain that are not valid.
- When calculating the reciprocal of a rational function, we must be careful to check the numerator of the divisor when identifying the domain.