Identify any vertical, horizontal, or oblique asymptotes in the g...

Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

Accepted Answer

Everyone in this problem for the graph of the rational function given below, we're asked to identify any vertical, horizontal or oblique asymptotes and also to write the domain of the function. So we're given a graph here with our function drawn in red. And we're gonna come back to that graph in just a minute as we work through this problem. Now, we're given four answer choices for this problem. Option A and B have just a horizontal Asymptote at Y equals negative two and they differ in the domain. OK. So option A has a domain from negative infinity all the way up to positive infinity. But in option B, we have a restriction on the domain. OK. Option C and D both have horizontal and vertical asymptotes and they have the domain of negative infinity to positive infinity. OK. They just differ in what those asymptotes are. So let's move up to our graph again and we're gonna start with our vertical as to OK. So a vertical Asymptote is gonna occur. We're gonna have like a some vertical line on our graph where it seems like the function is approaching positive and negative infinity as X gets closer and closer to this line. I mean, when we look at our graph, we don't see anything like that. We don't have our function jumping up to positive or negative infinity. We don't have any vertical line where it looks like the graph is not touching. And so based off of this graph, we actually have no vertical as to. So we can already eliminate some of these answer choices. Option C and D both had vertical ascent totes as part of the answer choice. And so we can eliminate those two options because we know there is no vertical as tool back to our graph and moving on to horizontal ascent tode. OK. A horizontal ascent tode is going to be a horizontal line that this function approaches as it gets really, really large. OK. So as this function goes off to positive and negative infinity or as the X values, I should say, go off to positive and negative infinity, this function is approaching what looks like a horizontal line and that horizontal line is located at Y equals negative two. And so that is gonna be our horizontal Asymptote Y is equal to negative two, which is the line that the function approaches as X gets really, really, really big, either positively or negatively. Now because we have a horizontal ascent to, we can't have an oblique ay to have a horizontal asom tope, we need the degree of the numerator equal to the degree of the denominator, OK. To have a horizontal asom tope that is not Y equals zero. In order to have an oblique Asymptote, we would need the degree of the numerator to be one more than the degree of the denominator. OK. So we can't satisfy both of those conditions. So we have this horizontal Asymptote which means that we have no oblique Asymptote. Now looking at the domain again, the domain is gonna be all of those X values that we can have any possible X value that we can substitute into this function. Now, we have no vertical asom codes, we have no holes shown in this function. There's nothing discontinuous about this function. And so the domain of this function is going to be all real numbers. OK. So we have the interval from negative infinity up to positive infinity. And if we compare this with their answer choices, we see that this will correspond with answer choice A, we have a horizontal Asymptote at Y equals negative two and the domain is from negative infinity up to positive infinity. Thanks everyone for watching. I hope this video helped see you in the next one.

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Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

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