Video Transcript
Determine the domain of the function 𝑛 of 𝑥 equals 𝑥 squared minus 𝑥 minus six over 𝑥 squared minus four divided by two 𝑥 minus six over 𝑥 squared minus four 𝑥 minus four.
Let’s begin by inspecting our function 𝑛 of 𝑥. We see that it is the combination of two functions that we’re going to define as 𝑓 of 𝑥 and 𝑔 of 𝑥. 𝑓 of 𝑥 is a rational function. It’s the quotient of two polynomials, as is 𝑔 of 𝑥. And we’re looking to find the domain of this function, where the domain is the set of possible inputs to the function 𝑛 of 𝑥. Well, when we’re working with a combination of functions, the domain is the set of inputs common to both those functions. There is a caveat, though, when we’re working with the quotient of two functions. In this case, the domain is the set of inputs common to both. But we have to exclude any of those that make the function that we’re dividing by, 𝑔, equal to zero.
So let’s look at the function 𝑓 of 𝑥 and the function 𝑔 of 𝑥 and first determine their domains. We said they were rational functions. And the domain of a rational function is in fact the set of real numbers minus the set containing any elements that make the denominator equal to zero. So we set that denominator equal to zero and solve for 𝑥 to find the values that we’re going to exclude. That is 𝑥 squared minus four equals zero.
To solve for 𝑥, we add four to both sides. And then we take the square root, remembering that we must take both the positive and negative square root of four. So the solutions to the equation 𝑥 squared minus four equals zero are two and negative two, meaning that the domain of 𝑓 of 𝑥 is the set of real numbers minus the set containing these two values of 𝑥.
Let’s repeat this with 𝑔 of 𝑥. Once again, it’s a rational function. So the domain is the set of real numbers excluding any values of 𝑥 that make the denominator zero. So let’s set the expression on the denominator, 𝑥 squared minus four 𝑥 minus four, equal to zero. To solve for 𝑥, we factor. And in fact we get 𝑥 minus two times 𝑥 minus two equals zero. For the product of these two expressions to be zero, we know that either one or other of the expressions must itself be equal to zero. But in fact, this expression is repeated, so we just need to solve the equation 𝑥 minus two equals zero. And when we do, we find that 𝑥 is equal to two. So the domain of 𝑔 is the set of real numbers minus the set containing the element two.
So by finding the domain of 𝑓 and 𝑔, we can identify the domain of 𝑛 of 𝑥. It’s going to be the set of inputs common to these two domains. But we still need to identify any values of 𝑥 that make 𝑔 of 𝑥 equal to zero. Now, of course, if a rational function is zero, it must be true that the numerator is zero. In other words, two 𝑥 minus six equals zero. If we then add six to both sides of this equation and divide by two, we find that 𝑥 equals three is the value of 𝑥 that makes 𝑔 of 𝑥 equal to zero. And so we found the domain of 𝑛 of 𝑥. It’s the intersection of the domains of 𝑓 and 𝑔 excluding that value of 𝑥 three that makes 𝑔 equal to zero. So it’s the set of real numbers minus the set containing negative two, two, and three.
