Question Video: Determining the Domain of Rational Functions Mathematics

Video Transcript

Identify the domain of 𝑛 of 𝑥 equals nine 𝑥 plus eight over three 𝑥 plus two.

Let’s begin by inspecting our function 𝑛 of 𝑥. Nine 𝑥 plus eight and three 𝑥 plus two are examples of polynomials. And when we have a function that is the quotient of two polynomials, in other words, one polynomial divided by another, we call that a rational function. So, let’s remind ourselves what we know about the domain of a rational function. The domain of a rational function is just the set of all real numbers. But we have to exclude any values of 𝑥 that make the denominator equal to zero. And this is because we don’t want to be put in a situation where we are dividing by zero.

To find any such values of 𝑥, the values of 𝑥 that we need to exclude from our domain, let’s set the denominator equal to zero and solve for 𝑥. That is, three 𝑥 plus two equals zero. We’re going to subtract two from both sides, and that gives us three 𝑥 equals negative two. Then, we need to divide through by three, which gives us 𝑥 equals negative two-thirds. This is the value of 𝑥 that satisfies the equation three 𝑥 plus two equals zero. And so this must be the value of 𝑥 that we have to exclude from the domain of our function. It makes the denominator zero. Since the rest of the domain is just the set of real numbers, we can represent this using set notation as shown. The domain of 𝑛 of 𝑥 is the set of real numbers minus the set containing negative two-thirds.