Simplify the function 𝑛 of 𝑥 equals three 𝑥 over 𝑥 plus eight plus six over 𝑥 plus eight, and determine its domain.
The first thing that we can notice here is that we’re trying to add two fractions together. And to add fractions, we must have a common denominator. This is true when we’re working with whole numbers. It’s also true when we’re working with polynomials or with variables in the denominator. In our case, our two fractions already have a common denominator. This means that we’re able to add the numerators. Three 𝑥 plus six just equals three 𝑥 plus six over 𝑥 plus eight.
Our question does want us to try and simplify this problem. So there’s one other thing we can do. We can notice that the three and the six both have a common factor of three. If we take out that common factor, we can rewrite three 𝑥 plus six to say three times 𝑥 plus two, all over 𝑥 plus eight. Since there’s nothing else that cancels out or can be simplified, this is the simplest form of our function.
Now we’ll need to determine its domain. Remember that our domain is all the possible 𝑥-values. We want to ask the question here, “Is there any value for 𝑥 that would not yield a valid result?” We should notice that we’re dealing with a fraction and we have a variable in our denominator. No fraction can ever have a denominator value of zero because we can’t divide by zero. We want to know for what value of 𝑥 with the denominator of this fraction be equal to zero. To isolate 𝑥, we’ll subtract eight from both sides, and then we’ll have 𝑥 equal to negative eight. What that means is we cannot plug in negative eight into our equation. Here’s what would happen. We would end up with three times negative six over zero, and that’s impossible. We cannot divide by zero.
What we can say about our domain is this: our domain can be all real numbers with the exception of negative eight, or all reals minus negative eight which would be the case for our function 𝑛 of 𝑥 equals three times 𝑥 plus two over 𝑥 plus eight.