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Quesjiol 1715055Aralyze the transcendenta cunve-'Note: (a) Type none if it is not applicable or there is none; (b) Do not put spaces in all vour answersIdentif...

Question

Quesjiol 1715055Aralyze the transcendenta cunve-'Note: (a) Type none if it is not applicable or there is none; (b) Do not put spaces in all vour answersIdentify the following:Domain: (xx2.x-intercept: (3.v-intercept: (0Symmetry with (chokes: Ted Yaxls; origin or none}:Horizontal Asymptote:Vertical Asymptote lanswer from least t0 greatest:Recions are divided by the following (answer from least t0 greatest):

Quesjiol 17 15055 Aralyze the transcendenta cunve- 'Note: (a) Type none if it is not applicable or there is none; (b) Do not put spaces in all vour answers Identify the following: Domain: (xx 2.x-intercept: ( 3.v-intercept: (0 Symmetry with (chokes: Ted Yaxls; origin or none}: Horizontal Asymptote: Vertical Asymptote lanswer from least t0 greatest: Recions are divided by the following (answer from least t0 greatest):



Answers

In Problems 27–32, use the graph shown to find $$ \begin{array}{ll}{\text { (a) The domain and range of each function }} & {\text { (b) The intercepts, if any }} \\ {\text { (d) Vertical asymptotes, if any }} & {\text { (e) Oblique asymptotes, if any }}\end{array} $$ $$ \text { (c) Horizontal asymptotes, if any } $$

Here we have given why equals two ethics equals two negative X Q minus seven Xs quit plus 16 x plus one went toe dueted with X X squared plus X plus turned eight. So this content is already defined. Now we have toe place This girl using a graphing calculator here is it is not a good thing we have to find first the X in line Disip So here this is the point I'm writing here. So this is minus fight and zero. And here this is four and Zito off Xing gossip and wind the steps up zero and four These are y in the X intercepts since the denominator doesn't have any real videos. That's why there is no what the girl s him towards. And now do you like the numerator with the denominator you will get Why? Because to something that with a question off public, a simple and now they domain and both are minus infinity to infinity, both domain and ranges also minus infinity to infinity. So this is our solution

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.

So for this problem, we are asked to find multiple pieces of information about the rational function that we are given. So the first thing is to find the domain which we can find by setting the denominator equal to zero. To figure out what it cannot be equal to now because X is squared here. It does not matter whether we have a negative zero, positive. There is no number that physically would make the denominator zero. So the domain is all real numbers. So we can write it as negative infinity to positive infinity. We are then asked to graph the function. So I graft this on my graphing calculator and I will upload the picture. This is generally what our function is going to look like when I plug it into the graphing calculator. Mhm. And we're just plugging this in exactly as it is into Y equals on the graphing calculator for letters. See, we're asked to find the vertical and the horizontal a central because there was no number that made the denominator equal to zero. That's how we find the vertical ascent. Tote. There are no vertical ascent. Oats horizontal. On the other hand, we compare the degrees of the numerator and the denominator in this case the degrees are the same remember degree as the largest exponent. So then the horizontal ascent it is Y equals a over B. Where A and B are the leading coefficients. So that would be why equals one over 10.25 Which when we divide gives us why equals four. So we have one horizontal A sento at y equals four for a letter D. When we sketch our graph, the only as I'm told that I can put in is the four which is going to go across right here. And then if we wanted to, we could plot a couple of points to kind of see the nature of the graph. So I could plug in zero, we know we're getting zero, which is going to be right here. Um I could plug in two, you know, I get four over two squared is four times 40.25 is going to be one, so I get to so to to and then same thing would follow over here with negative two. And we can see that the graph starts to make just like what we saw with the graphing calculator. And that is the answer to this particular.

We have the function f of X equals the co tangent of X, and the answer is the whole lot of questions about this function. So let's start by making a quick little sketch of what co tangent actually looks like. And the graph of co tangents ISS something like this uses the lie accesses of her glass. Um, tote has another vertical has until there and then we have another same looking portion the graph to another vertical as in tow. We have another portion here with the lie access is a vertical ASM toad on the negative side. And so on. The period of this graph and the space between the ASM totes is high units. All right, so the first question is the domain. Well, the domain are the X values that we're allowed to use. Well, we can use any angle we want except where we have a vertical as in total, so X equals zero X equals pi X equals two pi X equals negative pie, and so on are the values that we're not allowed to use. So the domain would be all the real numbers except interred your multiples Uh, bye. Zero times pi one time spy, two times pi negative one time spy, etcetera. Next they asked for the range or the Ranger all the y values that we have. So looking at the graph while these go up forever to posit infinity and down forever to negative infinity. So the range is going to be the interval from negative infinity to positive infinity or all real numbers. Continuity. Continuity is asking if it's continuous. So can you draw this graph without picking up your pencil on the domain values? And that's a key part of the definition. So this is continuous. Yes, it has continuity on the domain. Then where is it increasing or decreasing? Well, that's describing it as we go from left to right, so as an increasing or decreasing as we look at each of these portions of the graph. Well, when you're starting near and ASM Tote and as we do with right, the raft is going down. So this graph is going to be decreasing on each interval in our domain. Each of those portions of the graph it's increasing and then symmetry. If you look at symmetry, there's two types of symmetry that we talk about. We have even symmetry, which is symmetry with respect to the Y axis. Can you basically fold it on the Y axis and have it match up? And we have odd symmetry which is rotating the graph based on the origin and having it match up exactly with itself. So when we talk about it having symmetry, we obviously can't fold this and have it match up. But we can rotated 100 and 80 degrees around the origin. So this has odd symmetry, its symmetry with respect to the origin bounded nous, is there any values that limit how high this Congar? Oh, and the answer is no, because we're going up to positive infinity and down to negative infinity. So this is not bounded. All extreme have any extreme values. So that's related to the bound nous bounded nous. And since we are going up to infinity and down to negative infinity and were continuous all the way through, we don't have any maximums or minimums in any spots here on the graph. So there is no local or global extreme ASM totes. Yes, we definitely have ASM totes. There are lines that we're getting close to but not touching those ASM tones. O r X equals the integer multiples of pie just like we used in our description, Uh, the values we couldn't use in the domain. We also can describe this as okay times pi for K being an Inger. And then finally they ask about the end behavior of the graph. What's happening as X goes to positive, infinity and X goes to negative infinity. Well, because of this being periodic, there is no specific behavior that is happening here. So the limit as X goes to infinity and X goes to negative and friendly does not exist, So there is no definite describable and behave here.


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