Sketch a possible graph of a function g, together with vertica...

Question

Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions.

g(2) =1,g(5) =−1,lim x→4 g(x) =−∞,lim x→7^− g(x) =∞,lim x→7^+ g(x) =−∞

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Identify the graph satisfying the given conditions. F of 3 is equal to 1/2. F of 4 is equal to 0. The limit as X approaches 2 of F of X is equal to infinity. The limit as X approaches 5 from the left of F of X is equal to negative infinity. And the limit as X approaches 5 from the right of F of X is equal to infinity. Awesome. So it appears for this particular problem, all we're trying to do is we're trying to figure out the graph that satisfies these given conditions provided to us by the problem itself. So we're given Four different multiple choice answers A, B, C, and D for possible graphs or for any of these graphs could be our final answer. So we have 4 separate graphs, and as you can see for all 4 graphs, we have vertical asymptotes at 2 and 5, and these asymptotes are denoted as red dotted lines. And then we have curves in red. That of course, are within our vertical asymptotes. So once again, just in case you cannot remember, it's important to recall and note that these curves get closer and closer and closer to these asymptotes, but they do not touch and they do not intersect these asymptotes. So with that said, so we have, once again, we have 4 different graphs, and we're trying to figure out which of these 4 graphs is our final answer that satisfies these provided conditions set to us by the prom itself. So with that in mind, noting that our curves follow our vertical astodes, but they're all different, like, for example, for A. We have a curve that starts in the negative quadrant and then goes up into the positive quadrant, following our vertical asytope for 2. Then we have a parabolic curve in between our vertical asymptotes between 2 and 5 or X. So these are our vertical asymptotes. Once again, Which once again, these are for X equals 2 and X equals 5, we have like a parabolic shaped curve, and then finally we have a curve that goes in the negative quadrant and then works her way towards the X axis. As we can see, That's for A, and then for B, similarly, like to A, except it starts in the negative quadrant and then goes down to the negative quadrant again. So it's up really, really close to our X axis and then it dips down into the negative quadrant. Then we have an inverted upside down parabola in between our vertical and totes of 2 and 5, and then we have a curve that starts really, really high in the positive quadrant and then dips down. low towards the X axis, and then for C is similar. To the graph for B. However, instead of it being a a parabolic curve in between our vertical ascentopes, it's an elongated upright, almost like a cubic function. It's like an elongated S shape, but it's not like S looking, it's more like a cubic function. And then we have our third curve, which starts in the negative quadrant and then works its way up towards the positive. Or not positive per se, but it works its way towards the X axis. Awesome. And then similarly for D, we have a cubic looking function curve in between our vertical asymptotes, but instead of it going From down up, it's actually working its way from up to down. If that makes sense, hopefully. So, as we should note, so it's basically it's flipped. That's why I said that it should make sense because it's flipped. So with that said, now that we have a visualization of what's going on for our different multiple choice answers, let's see which graph will ultimately fit our provided conditions. So we need to know that we're told that F of 3. is equal to 1/2, and we're also told that F of 4 is equal to 0. So that will mean that our function F will pass through the points. 3, 1/2, which is following our XY formatting. So that will mean our function F will pass through 3. 1/2 and it will also pass through 4.0. So those are two coordinate, two different coordinate points that are curves for the function will pass through. And we need to know once again that as X approaches to. Our graph for F will approach, so this will mean that our graph for F will also approach positive infinity. Which implies that a vertical asymptote is at X equals 2. So let's make a note of that. So we have a vertical asymptote at X equals 2. Fantastic. And we also should note that we have a vertical asymptote at, like we were discussing, at X equals 5, because the function is approaching. Negative infinity from the left and approaching positive infinity. From the right, so negative infinity. For the left and positive infinity from the right. So I use L for left and R for right to keep things simple. Fantastic. So now by inspection, considering this information that we discussed together, when we inspect our graphs, the function, or rather the graph that basically supports all the given conditions given to us by the problem itself has to be the multiple choice answer letter C. Which as you can see C has to be our final answer because it's the only graph out of all of our multiple choice answers that fulfills the given conditions, and this is the one that starts from, from what it what it appears on our graph. So it's like close to our X axis at -2, and it follows our X-axis until it dips down, going in the negative quadrant, going past -5, going down following our vertical asymptote 2. And then in between our vertical isotopes for 2 and 5, we have our cubic function based curve. So it starts close to our vertical acetota 2 from the negative 5 direction and then it works its way up, then it curves, and then it crosses our x-axis. It looks at 4.0, and then it starts to curve and then go up and follow our vertical asymptote of 5 going up. And then looking at our vertical acetode of 5, on the other side, we have our last curve, which is very close to our vertical acetote, and then it curves slightly going towards our X-axis. At about 6, it starts at 0.60. It starts to kind of curve and then just follows our X-axe going to the right. And that's it. That is our final answer. Hooray, we did it. So once again, to quickly recap, our final answer has to be multiple choice answer letter C. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions.

g(2) =1,g(5) =−1,lim x→4 g(x) =−∞,lim x→7^− g(x) =∞,lim x→7^+ g(x) =−∞

Key Concepts

Vertical Asymptotes

Limits

Function Values

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