They asked us to analyze F of X equals Cosi kind of X, and they had a whole lot of things they want us to discuss. So let's look at the graph of this because the things that were going to be discussing our observed straight from the graph. So the graph of Cosi convex has a U shape with the lie axis being a vertical ASM toad on the left side. And then this value at X equals pi being a vertical as, um total on the right. And then we get the negative portion of the graph there again surrounded by some as, um, totes. And it keeps going and going and going and the same thing On the negative side, we have the negative portion of the graph with class in tow. Ahem, positive portion another vertical as, um told negative portion. And all of these positive portions have a value of one, and the negative portions have a value of negative. Long at is the bottom or the top of a graph. We'll get to that in a moment. These ASM totes are at by and to pie and three pie and so on. Going to the right and artist said the Y axis is a Nazem tote, and then that vertical Assam toad is a negative pie and negative two pi et cetera. So again, those values will be important, as we discussed the different things that they want us to mention as we're answering this question. So the first question is the domain. What values are we allowed to use for X? We can use any values for X, except for where we have the vertical ASM totes. So this would be all real numbers except zero one pie two pi three pie. Negative one pie negative two pi etcetera. So, except into juror multiples, I love pie. The range the range is asking what why values occur in our graph. And it's important that these all about amount at one and they go up forever. And these all have a peek at negative one and go down forever. So these negative portions of the graph are the interval negative infinity to negative one, including negative one. So the square bracket there and we're going to union that we're going to combine that with the positive portion which go from positive one against square bracket because these why values equal one two positive infinity. So there's the description of our range. Then they ask about it being continuous. Well, it's continuous if we condone it without picking up her pencil and on the portions where it has domain values. It is continuous. And that's the important part of the definition of continuity is we're only talking about it on the domain values. So, yes, this is continuous on the domain. We're not looking at those X values that are vertical as in coats. Okay, all right, The next question after continuity, they want to talk about where it's increasing and decreasing. Problem miss. Here we have so many portions of the graph, and some of it it is increasing and some of it is decreasing. So we have to describe both of these things. The decreasing and increasing adult is always described as you're going from left to right. So in each of these portions of the graph, half of it is increasing and half of it is decreasing. So let's look real quick at one of these portions because this is going from zero to pie. The halfway point is pi over, too And so if you look at all the positive portions, the left half is decreasing on the right half is increasing. So for the portions that are centered at by over two plus two Hi, Kay. The period of this is to pie. It's taken two PI units to cover an entire portion of the graph. So all of these low points, on the positive part, are all too pie units apart. So if we do two pi K, where Kay is an imager, it is decreasing on the left half and increasing on the right half. Now, for the negative portions, it's increasing on one side, decreasing on the other. And those portions. Let's look at this one that is halfway between pi and two pie. So this maximum value here is at three pie over, too. And then again, because of the period equaling two pie, all these other negative portions are going to be to pi times some manager units away. So for the portion centered at three pie over two plus to pi times K or case and manager, it is increasing on the left half and decreasing on the right half. All right, the next is symmetry. There are two types of symmetry that we're concerned about. There's even symmetry, which is symmetry with respect to the Y axis. Can you fold it? Oh, the Y axis. And if you look at this graph, we obviously cannot fold this on the Y axis. That portion would go right there into that open spot. So this does not have symmetry with respect to the Y axis. The other type of symmetry is respect to the origin. So can we rotate this 180 degrees? Well, if you turn this portion 180 degrees around the origin, it comes right on top of the negative portion there. And the similar thing happens to all these others. So this has symmetry with respect to the origin, which is often called odd symmetry bounded nous no. Are there any limiting values on this? Is there anything keeping it? That's a maximum value or a minimum value that we don't go above or below. And the answer is no because this is going to positive infinity and negative infinity. No, it is not bounded either above or below. Then they asked about extreme extremo are Max Mons and minimums. So obviously our positive portions have a minimum value. So we have a minimum. These were local extreme. Let me clarify that. Because there are global extreme over the entire graph. We have a minimum value of one at our by over two for the X value plus the two pi K to pie. Okay. Where Kay's and manager. And then on the negative portions of the graph, they have a local maximum of negative one at three pie over too. Plus the two I Okay, where Case and manager also. So those are local extreme. There are no global extreme. Next they asked about as and totes Well, there aren't any horizontal as in totes because this is being periodic and as you go to infinity were just repeating values over and over again. But we do have the vertical as in totes. And those were described back in our domain those air the interviewer multiples of pie. So I had by times k for an injured your K. And then finally the end behavior. What's happening as this goes to positive infinity and negative infinity What's what's the limit as we go out forever to the right and to the left. Well, because this is periodic, there is no specific and behavior. So the limit as we go to infinity does not exist. We're not approaching any specific value as we go out to positive in Trinity or Negative Infinity.