Explainer: Adding and Subtracting Rational Functions

In this explainer, we will learn how to add and subtract rational functions, identify the domains of the resulting functions, and simplify them.

First, we will recall the definition of a rational expression.

Definition: Rational Expression

It is a fraction with polynomials in the numerator, denominator, or both.

When adding or subtracting rational expressions, they behave the same as regular fractions, without variables. This means that we need the two fractions that we are adding or subtracting to have a common denominator. Once the two fractions have the same denominator, we sum the two numerators to obtain the numerator of the new fraction, and the new denominator is simply the common denominator.

Let us look at an example.

Example 1: Adding Rational Expressions with Common Denominators

Write 34𝑥+64𝑥 as a single fraction in its simplest form.

Answer

We start by looking at the denominator of each fraction in the expression, and we can see that in both cases it is 4𝑥. It may be tempting to simplify the second fraction here; however, if we did that then the fractions would no longer have a common denominator. Since the fractions have a common denominator, we can write them in one fraction to obtain 3+64𝑥. Then, we can simplify this to 94𝑥. Now we have written the expression as a single fraction.

Now, let us write the final expression from this example as the function 𝑛(𝑥)=94𝑥.

We can now find the domain of this function. The domain of a function is defined as follows.

Definition: Domain of a Function

It is the set of input values for which the function is defined.

For 𝑛(𝑥), we have only one variable that is input into the equation: 𝑥. The restriction here is that the denominator has to be nonzero, since we cannot divide by 0. The denominator of our fraction is 4𝑥, so we cannot input the value 𝑥=0. This gives a domain for our function of {0}.

Now, we will look at some more examples.

Example 2: Simplifying the Sum of Two Rational Functions and Determining Its Domain

Simplify the function 𝑛(𝑥)=1𝑥+38𝑥+3, and determine its domain.

Answer

First, we see that the fractions in the equation have common denominators, so we can write 𝑛(𝑥)=18𝑥+3. From here, we reach the simplified function 𝑛(𝑥)=7𝑥+3.

Now, we need to find the domain of 𝑛(𝑥). The denominator of the fraction is 𝑥+3, so the domain will contain all values, except when 𝑥+3=0. This is when 𝑥=3. Therefore, this gives us a domain of {3}.

Example 3: Simplifying and Determining the Domain of a Sum of Two Rational Functions

Simplify the function 𝑛(𝑥)=3𝑥𝑥+8+6𝑥+8, and determine its domain.

Answer

The two fractions here have a common denominator, so we can write 𝑛(𝑥) as 𝑛(𝑥)=3𝑥+6𝑥+8. Then, we factor the numerator to check if the numerator and denominator have a common factor 𝑛(𝑥)=3(𝑥+2)𝑥+8. Since there is no common factor here, this is the simplified function. To find the domain, we find the values of 𝑥 for which the denominator is zero. This is just when 𝑥+8=0, which gives us the value 𝑥=8. Therefore, 𝑥=8 cannot be an input in our function since this will give us an undefined value. From this, we obtain a domain of {8}.

Example 4: Simplifying a Function with Rational Expressions and Finding Its Domain

Simplify the function 𝑛(𝑥)=2𝑥8+48𝑥, and determine its domain.

Answer

First, we notice that the fractions do not have a common denominator. However, the denominators are in fact the negative of one another. In order to get a common denominator, we can multiply the numerator and denominator of the second fraction by 1. This will give us 𝑛(𝑥)=2𝑥8+(1)4(1)(8𝑥).Next, we expand the brackets on the fraction on the right and get 𝑛(𝑥)=2𝑥8+4𝑥8.

Writing this as one fraction, we obtain 𝑛(𝑥)=24𝑥8. Now, we simplify the numerator to get 𝑛(𝑥)=2𝑥8. All that remains is to find the domain of this function. This is all values of 𝑥, except when the denominator is 0, so when 𝑥8=0, this rearranges to 𝑥=8. Therefore, our domain is {8}.

Example 5: Simplifying and Determining the Domain of a Sum of Two Rational Functions

Simplify the function 𝑛(𝑥)=7𝑥𝑥1+3𝑥1𝑥, and determine its domain.

Answer

First, we notice that we do not have a common denominator. We need to make the denominators the same, so we can add the fractions. One way of doing this is to multiply the numerator and denominator of the second fraction by 1. This gives us 𝑛(𝑥)=7𝑥𝑥1+3𝑥𝑥1. Now, we have a common denominator and we can add the fractions, giving us 𝑛(𝑥)=7𝑥3𝑥𝑥1.

We factor the numerator to check if there are any common factors in the numerator and denominator. This gives us 𝑛(𝑥)=𝑥(7𝑥3)𝑥1. Here, we have simplified the function as much as we can. Now we will find the domain. We need to exclude 𝑥 values that give a zero denominator. This happens when 𝑥1=0, which is equivalent to 𝑥=1. Finally, we have a domain of {1}.

We will finish by recapping some key points.

Key Points

  1. A rational expression is a fraction with polynomials in the numerator, denominator, or both.
  2. When two rational expressions have a common denominator, we can add or subtract them by adding or subtracting their numerators to get the numerator of the result; the denominator of the result is simply the common denominator of the two rational expressions.
  3. The set of input values for which the function is defined are all the values for which the denominator is nonzero. Hence, given a rational function 𝑓(𝑥)𝑔(𝑥), where the zeros of 𝑔 are the set 𝐴, the domain for the function is 𝐴.

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