 Welcome back to our lecture series Math 1050, college algebra for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misildine. In this lecture, lecture 30, we're gonna start our fifth unit in this lecture series, unit five, which we'll call algebraic functions. So this will encompass other algebraic functions that we haven't yet talked about. We've done a lot about polynomials like linear quadratics and polynomials in general. In unit five here, we're gonna focus primarily on the idea of rational function, but also near the end of this unit, we'll talk about radical functions like square roots, q-brutes and such. But for rational functions, we define a rational function to be a ratio between polynomials. So if p and q are both polynomial functions, a rational function, which is where it gets its name as ratio, is gonna be the quotient of two polynomial functions. As such, a lot of the properties we've learned in the previous unit number four about polynomials will apply in this situation, but we have to also be concerned because of the division that comes into play. Polynomial division will be a big deal here. This affects the domain of a rational function. The domain of a rational function will be all those points that make the denominator not equal to zero. With the polynomial functions we've studied so far, there was no restriction on domains. So we have to worry about what makes the denominator go to zero. Now it turns out there's two things that happen to a function as the denominator goes to zero, a rational function. One option is gonna be something like the following. We get something called a removed point. So it looks like just a polynomial curve, but then when no one was looking, Bill will bag and stole the ring from Gollum and now it's missing. And so there's this removed point that one could see on a graph. It's just a point that is missing, okay? And that can happen. The other possibility is perhaps there's, what we call a vertical asymptote. Our graph kind of bends towards infinity or negative infinity as it approaches this value, this x-coordinate. So the functions, the functions at, the y-coordinate's absolute value approaches infinity. So it can be positive infinity or negative infinity at this point x equals c. So you get something like this, what we call a vertical asymptote right here. And so this is something we never saw for polynomial functions. So we wanna investigate what a vertical asymptotes for these rational functions. Now the good news is we can actually detect from the formula where a vertical asymptote's gonna be. So when you look at your polynomial here or this rational function p of q over q, p of x over q of x, right? This should be not reduced, don't reduce the function. So the not reduced, and I'll write this down here, that the domain here is determined by the non-reduced formula. And so we'll see some examples of this, the non-reduced fraction. We're often so used to like simplifying fractions by reducing them that we actually forget that some information can be lost when we reduce it. So if you wanna determine the domain of a rational function, do not reduce it. But when you find problems in the domain, when you find a place where the denominator goes to zero, these are what we call a discontinuity, okay? Discontinuity, because it's no longer continuous. Our graph cannot be drawn with some continuous stroke of our pen. The rational function's gonna have some break in it. And there's gonna be two types of discontinuities that we get in this situation. So the first option, like we saw in the previous slide, is the idea of a remove point. How do you get a remove point? Well, a remove point is gonna occur when you are in, when you do have a reduced fraction. When it's reduced, what you're gonna see with a remove point is that the discontinuity disappears, like in reduced form, q of x no longer is zero. But on the other hand, you get a vertical asymptote. You get a vertical asymptote, which when it's in reduced form, the denominator is still zero. And so let me explain to you via some examples. Sometimes it's better to explain things with examples here. So if you wanna determine the domain of this rational function right here, r of x equals x squared minus four over x plus four. In terms of the domain, the numerator means nothing. The numerator never affects the domain. The domain is gonna come from the denominator, for which if we set that equal to zero and solve it, we see there's a problem when x equals negative five. So the domain of our function is going to equal all real numbers x, such that x does not equal negative five. Or if we put this in interval notation, we want negative infinity up to negative five, union negative five to infinity. The value itself is outside the domain. It's an exception to domain. The numerator is what determines the domain. Has nothing to do with the denominator, excuse me, determines the domain. The numerator has nothing to do with it. Now, when it comes to, do we have a vertical asymptote or a remove point, now we have to look at the numerator. How do these things affect each other? Now this polynomial right here, if you were to factor the numerator, you could take out a two, then you get x squared minus two, which could then factor as, well you get x minus the square root of two and x plus the square root of two. This all sits above x plus five and x plus five. You can see that in this situation, the rational function doesn't reduce anymore. So it's in what we call lowest terms already. And so in lowest terms, you get this x plus five. So this discontinuity occurs in lowest terms. So this x minus five right here is actually indicative of a vertical asymptote on the graph of the function. So you have to reduce the fraction to see whether it's a vertical asymptote or not. Let's look at the next example. For this next example right here, you have x squared minus four in the denominator. So we have to figure out when does x minus four go to zero. Factoring this thing, you're gonna get x minus two, x plus two as a difference of squares. And so your problems are gonna be at x plus or minus two. In other words, your domain is gonna be everything except for two and negative two. But again, when you look at this fraction, the numerators of one, the denominator is x squared minus four. This denominator or this fraction is already in lowest terms. So there's no simplification that's necessary. This indicates that both x equals negative two and x equals positive two. These are both vertical asymptotes on the graph. Well, we'll talk more about exactly what the graph looks like later on. But what we do know already is there's some type of infinite rip on the graph. Our function is gonna kind of, as it gets closer to this asymptote is gonna be bending towards infinity and negative infinity, something like that. That's the behavior we can anticipate as we graph these in the future. We'll also get this function right here, x cubed over x squared plus one. To determine the domain, we just look at the denominator. And we solve the equation x squared plus one that equals zero. Well, you could try to factor it or you could add subtract one from both sides, x squared equals negative one, take the square root x equals the square root of negative one that is plus or minus i. When it comes to the domain, by the domain convention, we throw out any real numbers that make the quotient be not a real number. For graphing these things, we're actually not gonna allow imaginary numbers here. So in fact, the domain in this situation is going to be all real numbers. There is no real number for which makes the denominator go to zero. So as such, there's gonna be no vertical asymptotes on this thing. Because there is no problem with the domain, there's gonna be no vertical asymptotes. And that's a possibility here. Looking at example D, you're actually gonna see a very similar thing, right? This is technically speaking a rational function because the denominator is just a constant polynomial. When does three equals zero? Well, it doesn't. It's not possible to make three equals zero. Which admittedly, this is just a polynomial, right? One negative one third x squared plus two thirds. This is a polynomial, but every polynomial is a rational function. But in conclusion here, the domain is gonna be all real numbers. And just like we observed on the previous example, there's gonna be no vertical asymptotes on this graph. Polinomials don't have vertical asymptotes. All right. Now let's get to the real heart of the matter here. Let's take the rational function r of x this time to be x squared minus one over x minus one. If we investigate the domain, we only look at the denominator of the original expression. And we see that the domain is gonna be all real numbers x except for one, right? When x is one, that makes the denominator go to zero. But when you factor out the numerator, right? x squared minus one, that's the difference of squares. You get x plus one and x minus one. You then see that the x minus one cancels and in lowest terms, we get the polynomial x plus one. Notice that in this consideration, the x actually could equal one in this situation. There's no division by zero anymore. And so what that means for us is that when it comes to x equals one, this is actually a removed point. Notice how the discontinuity disappears when you simplify the fraction. And so in terms of the graph, we're gonna see a point that's missing, not that infinite rip associated to a vertical asymptote. Let's do one more example to illustrate the difference here. And so if we take the rational function x squared minus nine over x squared plus four x minus 21, the domain is only affected by the denominator. You never need the numerator for the domain. So when you factor the denominator, you're gonna get x plus seven and x minus three. So that tells me that the domain is gonna be all real numbers, except it's easier just to write the exceptions, negative seven and three. So the function's not to find a negative seven and three, but what happens to the graph? Well, when you look at the, when you factor the numerator, x squared minus nine, you get x minus three and x plus three. You see the x minus threes cancel, but the x plus seven didn't cancel out with anything. So what this tells me is that x equals negative seven actually coincides with a vertical asymptote because it didn't cancel out like the x minus three does. On the other hand, x equals three, this is gonna be a removed point. And so that's the difference. The domain will be anything that makes the original fraction go to zero. That is, those are gonna be the things outside the domain. Then in terms of the graph, we have a vertical asymptote, this infinite rip, or just a missing point. That depends on when we cancel things out. If stuff in the denominator cancels out, it was just a removed point. If it doesn't cancel out, then it was in fact a vertical asymptote.