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Find the following for the function; g(*) =_ X-9 Y-81(7 pts )a) Domain:b) y-intercept: _equation of vertical asymptotels): _d) equation of horizonta= asymptote:...

Question

Find the following for the function; g(*) =_ X-9 Y-81(7 pts )a) Domain:b) y-intercept: _equation of vertical asymptotels): _d) equation of horizonta= asymptote:

Find the following for the function; g(*) =_ X-9 Y-81 (7 pts ) a) Domain: b) y-intercept: _ equation of vertical asymptotels): _ d) equation of horizonta= asymptote:



Answers

Sketch the graph of each function and find (a) the $y$ -intercept; (b) the domain and range; (c) the horizontal asymptote;and (d) the behavior of the function as $x$ approaches$\pm \infty .$ $$f(x)=7 e^{x}$$

We guys in this problem were given X minus seven over X squared minus 49 asked to find the values of expert which there are vertical assam totes. So before we do this, the normal way to do this is if you do X minus seven over X squared minus 49. On that, you're basically gonna have todo basically set the denominator equals here. But before we set the denominator equal to zero, we can do some factoring out of the way we can do some factoring out to reduce their problems. So fact around and simplifying this explosion. Now, if you do explain seven, the denominator is gonna be X minus seven times Expo seven, the X minus seven cancel out. So whatever things cancel out, there is no vertical ascent. Oh, but you know that there's a hole at that point. There's a hole at that point, So basically they're at X equal. Seven. There's a hole, but there is no medical ass in tow. Now you're left here with one over X plus seven and when is one over X plus seven Undefined When experts seven is equal to zero. So for value X equals negative seven. So this is gonna be our answer to the only value for which you're gonna have a vertical ascent. Oat is a negative seven. But keep in mind that there is a whole point x equal said that that's the point that factors

Were given a rational function. This is G F X equals X cubed over two X squared minus eight. There's so many. Yeah, In part, they were asked to take the domain of this function. Notice the function is equal to an FX or VFX where n FX sex cute and dfx is two X squared minus eight and n and d of no common factors. Jeez, the domain. These are all real ex. Except for those for which d of X equals zero or two X squared minus eight equals zero. No, she so X equals plus or minus two. Therefore, the domain, it's all is all real X except for X equals plus or minus two. Oh, there. Uh huh. Also in part B rest to identify all, uh, intercepts. Mm. Zero. And we now to find the intercepts, the X intercepts. These are the zeros of the function. And these are the same as the zeros of n or solutions to N of X equals zero. So zero equals execute So that X equals zero. It's not like sorry. Therefore, we have the X intercept. Well, he seeks to 00 This also happens to be the y intercept. Yeah, as you can verify by plugging in zero. Yes. Yeah. No courses in part C. We're asked to find any vertical or slant. Asthma totes the vertical asthma totes. These are the values of X for which the the denominator DMX zero. This is the same as part A. So we have vertical assim Toots X equals plus or minus two, That's what. Yeah. Now the degree of the numerator n this is three, which is two plus one, which is the degree of the denominator d plus one. Since the degree of the numerator is exactly one more than the degree of the denominator is slant asthma. Toad exists to find out We're going to use long division. So g of X is equal to X cubed over two X squared minus eight. Right. First, I'm going to factor out a one half so I have X cubed or one half times X cubed over X squared minus four. Yeah. Mhm. Now to divide this execute divided by X squared minus four. First I multiply X squared minus four by X fits an x cubed that many times I'm going to subtract from this x cubed minus four X I get execute minus X cubed and then positive. Four X x squared minus four Does not fit into four X. So we end up with X plus a remainder of four X over X squared minus four. Come side you're on. Now. The polynomial term in this see equation is our slant. Asthma tote. So we have a slant. Assume toot y equals X jack, then Empire D rest additional solution points as needed. To sketch the graph, make a table with two rows X and G f X. Now, in the first table, I'm going to choose values of X that lie between the X intercepts and the vertical Assam totes, for example. So there. Oh, you know what? I made a mistake here. So this is actually one half times x times four x over expert minus four. And so the stank attitude is actually y equals X over two. Okay, okay. And now we have vertical attitudes plus minus two and x intercept at zero. So, yeah, I'm going to choose X to be negative. Three negative. One, one and three. Good. Now G of negative three. This is negative. Three cubed or negative. 27 over two times nine or 18 minus eight. This is negative. 27 over 10. Somebody stop G of negative one. This is negative. 1/2, minus eight. Negative. One of her negative six, which is just 1/6 G of one. This is one over negative. Six or negative. 1/6 and G of three. 27/18. Minus eight or 27 10th. You know, because they do that. Any believes that? Now I'll plot the function. He's bad here too. Just have Yeah, there's something here. Decide every Yeah, yeah, yeah. Sponsored. Yeah, I'm really, really some. Yeah. Now I smoke weed with the police. The X intercept allies in the graph at the origin we have vertical ascent totes that X equals plus or minus two respect. Mhm. But yeah. Can't remember if it was your brother, Was it lander lfo then the slant asientos y equals X over two Says appointed the origin like and as a slope of one half. So appoint at 21 as well. Do sure Chinese people B six. They're the ones. Yes. Notice that there is actually a point on this aspect to it. which lies in the graph. So we called it asthma totes. They really only determine the end behavior of graphs. Finally, I'm gonna start plotting these points so we have negative three negative. 27 10th. It's a little bit greater than negative. Three negative three. So it's about here still underneath the asthma tote. Then I have negative 1 1/6 which is about here above the asthma tote. Then one negative 1/6 which is below the acid toot than 3 27 10th which is just above the ascent coat. And yes, so we draw the graph the function by starting at negative infinity following the slant Assam tote until we turn around and go back towards negative infinity following the vertical ascent oot. Then we begin a positive infinity. Follow the vertical ascent tote until you reach the origin, we sort of flatten out. Then we start heading towards positive. Sorry, negative infinity and follow the other vertical ascent oats through again. Oh Then we start at positive infinity again following the vertical ascent tote until we turn around and start following the slant Assam tote back towards positive infinity. So the graph of our rational function looks something like this

Given the function F of x equals four X minus nine over x squared minus nine. Yeah. We first want to identify the X intercepts the X intercepts are when the numerator is going to be zero, when that's when the whole function will be zero. And so when is four X minus not going to be equal to zero, That would be at 904. So X equals 9/4 should be um an X intercept Vertical Aspen Toaster. When the denominator is equal to zero so that's going to be A positive and negative three. Those are going to be vertical ass photos because X cannot be either of these otherwise re dividing by zero. Since the degree is larger in the denominator why equals zero will be a horizontal asuntos? And the y intercept is when um X zero so that before time zero minus nine or negative 9/0 squared minus nine which is negative nine which is one. Yeah.

Or given a rational functions. G f X equals X squared plus one over X was in part a whereas to state the domain of this function, these are values of X for which the denominator we call it D F X is equal to zero. This is one X Z equal to zero. So it follows that the domain is going to be all real x. Yeah, except X equals zero stars don't seem Jack Nicholson and part B rest to identify all intercepts. First, we find the X intercepts by finding when NX equals zero or zero is equal to x squared plus one. However, this equation has no real solutions and therefore there are no X intercepts. It's something said yes. Find the whiner steps you want to find g of zero. However zero is not in the domain. Therefore, there are no y intercepts either. Why are you fucking chill? Like so? In fact, do you guys have sex in the sixties? There are no intercepts in general. Okay, then in part c right. We're asked to find the vertical and slant Assen totes The vertical ascent totes. These are again the values of X for which the denominator d of X equals zero. So by party, we have the vertical assume Tote X equals zero like on to use that were turquoise and silver. Fucking Georgia O'Keefe is Santa Fe style onto its like like then find these slant asthma totes. Well, first of all, notice that the degree of the numerator is to which is equal to one plus one. And the degree of the denominator is one. This is the degree of D plus one since the degree of the nominator is exactly one more than the degree of the denominator, the numerator since the degree of the numerator is exactly one more than the degree of the denominator. It follows that a slant asientos exists. We can find it by using synthetic division or long division. So our polynomial function rational function G fx is also equal to If we divide through by x x plus one of her ex The polynomial portion polynomial term is our slant Astute. So we have a slant Assume toot make yes y equals x Yeah, higher. It's higher experience. No, it's a museum. Then in part d I just came to be ready, were asked additional solution points as needed to sketch the graph. So we make a table of values First, Roby X second road G of X. I'm going to choose values of X that lie between the vertical ass and totes and the X intercepts. Now we have no X intercepts and we only have one vertical ass and totes. I'm actually gonna take a few more excess. So, for example, negative to negative one positive one in positive, too Mhm Now G is negative. Two, this is four plus one over negative two were negative. Five has g of negative one is one plus one over negative one or negative too G of one is one plus 1/1 or two and G of two is four plus 1/2 or five halves. Good like always no city. Now I'll sketch the graph of the rational function. So first a plot. The vertical institute This is just X equals zero. There's smart for how he there's a different set. Mhm. We just have certain like then I'll plot these extra points. Which is this white case? Axes like a disease. You just get what from? Thanks really actually plot the slant assented first. Yeah, Stone makes you argue. I was the first one to save. So our slant has some tote is y equals X. It looks something like this. Puts on my headdress and says his for girls. Well, it is now, now a plot. The points we have pointed negative to negative five halves or negative. 2.5. So it's about here. It's kind of hard to see. We also have a point at negative one negative two here a point at 12 about here and to five halves about here. So I'm going to begin at Negative infinity and follow our ascent, Oats or slant. Assume tote until I head back towards the vertical ascent Tote. And I follow that. Then getting a positive infinity, I'm going to follow the vertical ascent it again until I follow the slant. Asientos back to positive infinity seven. Yeah, God was pulling. Frankly, he sh


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