 Welcome back to our lecture series Math 1050, College Odsber for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. So in lecture 48, we're getting drawn to the near of our lecture series here. I want to review the idea of asymptotic behavior of a graph and why are we going to do that? Well, recall from the previous video that as we tried to analyze the range of various functions, we were somewhat restricted on how easily we could compute the range compared to, say like looking for domains. When it came to domains, we just have three problems we have to look out for, division by zero, square root of negatives, and logarithms of zero or negatives as well. So it's easy to do domain, but range is much more tricky. I emphasize that we want to appeal to the graph of a function if we truly want to understand its range. So we use graph transformations amongst other techniques to try to understand how the graph of a function would look like. But we should mention that while graph transformations work for many functions that we've seen so far, there are many where graphs, I should say there are many whose graphs alone aren't going to be enough. This is going to be the case when we studied polynomials and rational graphs earlier in this lecture series. Just so many of them could not be expressed as transformations that is just shifting, stretching, and reflection alone wouldn't describe the graph. We graphed those functions, the polynomial of rational function that is. We graphed those functions by looking at the in-behavior, that is what happened as x towards infinity or negative infinity. We also asked what happened when the graph got closer to say like x intercepts and y intercepts, like did it touch the x-axis? Did it cross the x-axis? So all of this discussion right here is the idea of the asymptotic behavior of this graph. It turns out this strategy of looking at the asymptotics works very well for a large family of graphs. That is, we can extend what we did previously to so many other things, and that's what we're going to talk about here in lecture 48 in this video and in the following two videos as well. So what you now see on the screen here, let me read this for you. Recall that the in-behavior, what was the in-behavior again? It was the trend of the graph as x continues towards infinity or negative infinity. We often would write this as as x approaches infinity, or we'd say as x approaches negative infinity. But that was kind of assuming the domain of the function was negative infinity to infinity. It could be that you can't approach infinity or negative infinity because it's just outside the domain of the function. There could also be various discontinuities in the inside of the function. So like a prime example of this type of discussion here, like take y equals the natural log of x. Sure, you can ask what happens as x goes to infinity. You'd say that y goes to infinity as well. But when you start asking what happens as x goes to negative infinity, well, on the natural log graph, you don't get anywhere close to negative infinity. You can't do that. The boundary, the in-behavior is gonna be what happens as you approach zero from the right-hand side. And then we see that y approaches negative infinity in that situation. So when we ask about the in-behavior, we're asking what happens as we go to the extreme as you go to the far left to the far right of the graph. Okay? And so like I said, for many functions though, it turned out that we could approach both positive and negative infinity. Now, when we computed the in-behavior for polynomials, we saw that the leading term determined the in-behavior, right? That first term, I should say the biggest term with respect to the polynomial, it determined the in-behavior. And now the argument was that as the absolute value of x approached infinity, all the other terms were dwarfed in comparison to the leading term, right? Sure, x squared might grow rapidly, but x squared grows nothing compared to how rapid x cubed grows. But x cubed grows insignificantly compared to like x to the fourth or x to the fifth. And so this idea of looking at the leading term because the other terms become minuscule in comparison, this is to say that the leading term of the polynomial was dominant. This was the dominant term. As x went towards its in-behavior, this one term became so much dominant over all the others that the other ones were recessive in comparison. Sometimes this is called the asymptotic term because we'd often say things like your polynomial f of x was approximately equal to say like x to the n as x went to infinity, right? So x would be asymptotically the same thing as its leading term. So the graph will have the same in-behavior as its dominant term. And so for sums of power functions, such as polynomials, right? So power functions in general, your exponent could be any real number. With polynomials, we only allow it to be positive integers. I guess we could have a zero as well. That's a constant term. But for sums of power functions, the dominant terms always give you the largest power. So even if those exponents were fractions, even if they're irrational, even if they're negative, the dominant term is gonna be determined by the biggest power presence. So that makes it pretty easy for determining dominance for power functions. That is sums of power functions. You just look for the biggest term, right? For which you can see an example of that on the screen right now. I wanna point this out to you. So if you look at this function, f of x equals five x to the fourth minus three x plus seven x squared minus two x to the eighth here, the dominant term is gonna be the power function in this sum that has the highest degree, which of course is gonna be this negative two x to the eighth. So what we see here is that as x approaches plus or minus infinity, our function f of x here will be approximately the same thing as negative two x to the eighth. Now the advantage here is that this is a monomial. We can describe its behavior using transformations. So you'll notice that there's a negative sign in here. So if you started off with x to the eighth, right? If you take like y equals x to the eighth, this is an even monomial. So it's gonna have this bucket shape, right? Where you get flatter near the origin, the bigger the power is, and you get steeper away from one and negative one that the bigger the power is. So we're gonna get, just intuitively here, we're gonna get a picture that looks something like the following, right? Again, it doesn't actually have these sharp corners. This is just to give us an intuition of what happens. So then when you look at y equals negative x to the eighth, right, you can even throw in the negative two there, x to the eighth, because the tools vertically stretch the graph, but in terms of in behavior, that stretch doesn't mean anything. The reflection matters. So when you reflect this thing, your bucket's gonna be going downward. So we can see the in behavior here. f of x is going to approach infinity as x approaches infinity, we see that. But we also see that as x approaches negative infinity, f of x here will approach positive, or excuse me, these should both be negative infinity because it's pointing downward. Let me correct that. Negative infinity and negative infinity because we see on our picture that it is pointing downward like so. So we can see the in behavior because the in behavior of f will mimic this power function, this monomial, which was the biggest power present there. So if you have some type of like sum, sum or difference, these combinations of power functions, it's easy to determine dominance. When exponentials and logarithms get into play though, it becomes a little bit more subtle and I wanna pay attention to some of those examples right now. So when you have an exponential function, it turns out that exponential functions in general are more dominant over power functions when they're growing, right? When you have a growing exponential function, it's gonna be more dominant than any power function whatsoever. But on the other hand, exponential functions are gonna be recessive to power functions when they're decaying. What I mean is like, if you think about these options, like one growing exponential could look something like this and then a decaying one might look something like this over time, okay? For which you'll notice that when you decay, you're going off towards some horizontal asymptote. Clearly the power function is gonna go faster than that type of decay, but compared to the growth of this exponential, the power function can't compare to it. So exponentials dominate all power functions when they're growing and they're recessive to all power functions when they're decaying. Logarithms are gonna of course do the exact opposite because they're the inverses of exponentials. When you have a logarithm, they're recessive to power functions when they're growing. So like a power, like a logarithm would look something like this, right? Very slow growth over time. That'll be recessive to a power function. So on the other hand, they're dominant to a power function when they decay. So a decaying logarithm would look something like this, maybe. And so we would anticipate, you know, we can anticipate this type of dominance happening in that situation. So you have to be really careful, of course, when it comes to logarithms, what's going on here. So we have to pay attention to the in behavior of these things and we'll give you some specific examples to make very clear what is we're doing. So we saw this example A already. Let's look at another one. So looking at example B right here, we have G of X equals X to the one-third minus three X plus two times X to the five-thirds. You'll notice that in this example, everything is a power function. When it's only power functions, dominance is very easy. You just look for the biggest power and that biggest power is gonna be X to the five-thirds. So as X, as the absolute value of X approaches infinity, we see that G of X will be approximately the same thing as two X to the five-thirds. So what does that graph look like? What does it mean to graph? You know, because the previous one was a monomial. We've learned about monomials before. What does it mean for X to approach? You know, how do you graph X to the five-thirds here? Well, one thing we've seen previously that when you have a power function, right, Y equals X to the N, as this number N gets bigger, bigger, bigger, like if we just look at what happens to the right-hand side of the Y axis, this is our Y axis right there. So notice that, and then we'll have an X axis there as well. As you graph this thing, as N gets bigger, this thing is gonna get steeper and steeper and steeper and steeper. There's no exception to that. There's no exception just because we have fractions here. And so when you look at something like five-thirds, if you wanna think about how steep it should be, well, five-thirds, it's less than six-thirds, right? Which of course is two, but it's also bigger than three-thirds, which is one. So if we draw X equals X to the one, that's just a straight line like so. If you draw X squared, that's gonna be a parabola. So you get something like this. So our function X to the five-thirds, it'll be somewhere in between, kind of close to the parabola. Y equals X to the five-thirds. In particular, what we care about with N behavior as X approaches infinity, we will see that G of X will approach infinity as well. You'll notice that this is a positive two in front, so it does stretch it by a factor of two, but a positive stretch doesn't affect the N behavior. It's only a negative reflection that we really care about in terms of N behavior there. Well, what happens on the right-hand side? Well, this is where it gets a little bit more subtle. The fraction does make a big difference in that regard. So if we look at the denominator for a moment, that matters here. When we have the denominator, which is here, it's gonna be one-third, right? It's a cubic function, a cube root, for which there's no domain restriction, right? We're not taking, we can go to the left-hand side of the Y axis, because as we take negatives, we can take the cube root of a negative. There's no worry about domains, so we do have something happening over here. If this was like five-halves, then this would be the picture, right? There would be no more picture to put there. There would be no left-hand side because of the domain, all right? And so that would mean that you couldn't approach negative infinity. You could only approach zero from the right, okay? That's an issue we might approach in a different example here. But so this one being the cube root means it's okay. We can do something on the left. But what about the numerator here, right? So the numerator's gonna be this five-thirds, and so that's also an odd power. Like if it was an even power, it means it's gonna be going back up. That is, you're gonna be going in both the same direction, right? But since it's odd, we actually see that it's gonna be doing something like the following. Again, the exact steepness that you don't have to worry about. It'll kind of, this will be an odd function, right? If you put in negative one, it'll stick out on negative one. So it should be symmetric with respect to, with respect to the origin. And so we see that as x approaches negative infinity, g of x will approach negative infinity as well. So we see that. Because again, we're not graphing g of x right now. We're thinking of what is the graph of two x to the five-thirds look like without the use of our calculator. We can force to help ourselves with the calculator, but we don't need it necessarily. We can determine how this power function behaves from what we've done previously. And then that'll determine the in behavior of g as well. All right. And so in this, in this example, c, notice we scan through it. We wanna find the in behavior of this. Who's the dominant term? Scan, scan, scan, scan. We see again, these are all going to be, these are all power functions. So the biggest power is gonna determine the in behaviors. The dominant term here is x squared. So as x approaches infinity, we see that h of x will approach x squared. It'll be approximately the same thing I should say. So in particular, h of x is going to approach infinity. And so if we think of the graph of what's going on here, you're gonna get y equals x squared. You get something like this, right? Y equals x squared. That's the dominant term. And so as x goes towards infinity, these other terms become minuscule in comparison. Even the three halves, right? Three halves in terms of a fraction is kind of close to, right? But it doesn't matter. In the long haul, even x to the three halves will become insignificant in comparison. So that's what the, that's kind of what the right hand side of the graph is gonna look like. But like I was mentioning on the previous slide, we do have to be cautious about the domain of this thing. As we look through these power functions, are there any pieces which have restricted domains? When we look at x squared, x squared's gonna be defined for all real numbers. So will three x and so will x to the one third. The cube root has no restriction there. But when you look at x to the three halves, x to the three halves, because the denominator is two, this actually looks like to us negative three halves. This is the same thing as negative the square root of x cubed. And, or you could also think of as negative, negative the square root of x cubed, whichever you prefer. It's gonna be the same in this situation. So we see here that because you're taking a square root, this actually restricts the domain of f. The other four, the other three terms, x squared, negative three x and two x to the one third, they have no restriction on domains. But that x to the three halves, because it involves a square root, it does have a restriction. So the domain of this function is actually going to be zero to infinity. So we see that we actually cannot approach negative infinity. So the in behavior we need to be asking on the left hand side is what happens as x approaches zero from the right. What happens there, okay? And so this is something, this is where it can get a little bit fishy here, right? We're gonna see that it's gonna go towards zero, right? In a later video, we'll talk about how we approach zero. We could be a little bit more subtle about it. But you notice here that if you just plug in zero for all these terms, this thing is gonna go towards zero. So we see that here h is gonna approach zero as well. So we get something, well, we'll go to zero on the left and then we can go to infinity on the right hand side. Again, we'll improve our approach to zero in just in a later video. I don't wanna delve into it too into yet. At the moment, I just wanna kind of determine this idea of dominance. And so one last example I wanna mention for this video right here is what if we have the function f of x equals x to the eighth plus two to the x plus one tenth to the x. You'll notice that in this situation unlike the previous ones, there are some terms which are not power functions. We have an exponential function two to the x right there. And we have another exponential function one over 10 to the x. Now, you'll notice that in this regard we have a growth one, right? As x gets bigger, two to the x is gonna get bigger. But as x gets bigger, one over 10 to the x is actually gonna get smaller. So this is a decay exponential. And so their dominance depends on which direction we're going. So if we ask ourselves as x approaches infinity, who is dominant in that situation? And that situation, the dominance will then go towards two to the x. So what I mean is kind of like the following. If you think of all these graphs, right? Let's just kind of put it right here. We'll call x the eighth of the blue one. x the eighth, as we go towards infinity it's a power function. So it'll grow something like this. The exponential will grow faster, much faster. And then the other exponential because it's decaying it'll be going off towards zero, right? This thing, as x goes to infinity, this decay model will go off towards zero. So it basically offers nothing in the long haul. Yes, x to the eighth will go towards infinity. Two to the x will go towards infinity, right? So we can make that a little statement here. This one is going off towards, we'll just say it's going off towards zero. x to the eighth is going off towards infinity. Two to the x is going off towards infinity. But this is what we were saying at the beginning of the video. Exponentials grow much faster than a power function. So in the end, a growing exponential will dominate the power function. So we see that f of x will be approximately two to the x as x goes towards infinity. Now it's still true that f of x will approach, it'll still approach infinity here but we wanna specify like how quickly is approaching infinity, it'll look like two to the x. Now, if we flip our direction here we ask what happens as x goes to negative infinity. Notice negative infinity is no problem with the domain here, right? These functions can approach negative infinity unlike the previous example. As x approaches negative infinity who's dominant this time? Well, with our power function, with our power function x to the eighth right here you'll see that it's even. So it's gonna be reflective across the y axis, symmetric to the y axis, excuse me. So it'll also be approaching infinity. If we look now at two to the x it's actually gonna be decaying, right? It's gonna be decaying towards zero right there. So on the left hand side it's basically gonna be in a skew. But on the other hand, the one over 10 to the x it's gonna be rapidly, it's gonna be going up and up and up and up and up and it's going towards infinity. And so dominance will be given to the decay graph as we go towards negative infinity. And so in summary there we would say that f of x will be approaching, excuse me it'll be approximately the same thing as one over 10 to the x. And so yeah, we see, we see in fact that y is gonna approach infinity still. So it points up and up on the left hand side but to the dominance like the baton is passed from the growing exponential from the right hand side to the decay exponential on the left hand side. Because after all, if you think about it in terms of reflection, if you reflect a decay model it becomes a growth model and growth becomes decay. And so as we approach infinity it's kind of obvious that the growing one beats the power function, the decaying one. But as we approach negative infinity it's kind of like we're going back in time. And so in terms of reflection the decay one is actually now the fastest growing if we rewind the tape in that situation. And so in summary here when it comes to dominance the biggest power will dominate power functions but exponentials gets a little bit more tricky, right? When the exponential is growing it'll dominate a power function but when it's decaying it'll be recessive. That's if you're going towards infinity if you go towards negative infinity you get the exact opposite.