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

Hey, everyone in this problem, we have a graph of a rational function and we're asked to identify any vertical, horizontal oral bleak asymptotes and also to write the domain of the function. So we're showing our graph of our function here and it's in blue. And you can see that there are three sort of sections to this function. Now, we're also given four answer choices A through D and they just contain different options for those ascent totes as well as the domain. And we're gonna come back to those as we work through this problem. Now, let's start with their asymptotes and let's start with their vertical asymptotes. OK? And recall that the vertical asy totes are going to be vertical lines where the function is approaching either positive or negative infinity. So we can see in our graph, we have this dashed red line that's drawn at X is equal to negative four. On the left of that line as our X values get closer and closer to X equals negative four. The function is approaching negative infinity. And on the right side of that line as the X values get closer and closer to negative four, the function is approaching positive infinity. And so that line is going to be a vertical as to, so that's gonna be at X is equal to negative four. Now, that's not the only one we can look again and we see another line that's similar at X equals positive four. And this graph looks fairly symmetrical. So at X equals positive four, same thing on the left hand side, the function is approaching positive infinity as the X values get closer and closer to four. And from the right hand side, the function is approaching negative infinity. And so we have another vertical Asymptote there. So our vertical asymptotes are at X equals negative four and X equals positive four. Moving to our horizontal lays. Our horizontal asymptotes are going to be horizontal lines that the function is approaching as the X values get really, really big in either direction, either negative infinity or positive infinity. We can see in our graph, we have this dashed purple line as X goes to positive infinity. Our function is approaching that dashed purple line and as X goes to negative infinity, the function is also approaching that same dashed purple line. So this is gonna be a horizontal axent tote and that line is located at Y is equal to negative four. So we have a horizontal ascent to at Y is equal to negative four as well. Now, for oblique asymptotes, if we have a horizontal asy tote, that automatically means we don't have an oblique Asymptote. OK? Or horizontal asy tode that are nonzero. We need the degree of the numerator to be equal to the degree of the denominator of our rational function. For an oblique Asymptote, we'd need the degree of the numerator to be one bigger than the degree of the denominator. OK. So if we already have a horizontal Asymptote and our equation satisfies those conditions, then it can't satisfy the conditions of an oblique Asymptote. So we don't have to worry about that. In this case, we have our vertical ascent totes and we have our horizontal Asymptote. And now we're gonna look at the domain of our function and the domain of our function is gonna be related to our vertical asymptotes. You can see that we can have any X value for this function and we're gonna get a corresponding Y value. However, when we have our vertical ascent to the function is gonna be undefined there. So our domain is gonna be all X values except for X equals negative four and X equals positive four. So we're gonna have to write this in three intervals and we have the interval from negative infinity all the way up to four, oops all the way up to negative four, sorry, but not including it. A we use a round bracket to indicate that we don't include negative four because our function is not defined there. We take the union of that with the interval from negative four, up to positive four. Again, round brackets, we're not including the end points. And finally, we take the union of the interval from four up to positive infinity. OK. So what we're doing here is we're including all the possible X values in our domain, except for the ones at exactly negative four and exactly four. OK. So we've split our intervals up there and used round brackets to indicate that we are not including those points. So that is our domain. Now, if we look through our answer choices, OK, we can see that option A has only a horizontal as to, we can eliminate that because we know we have vertical accent totes as well. Option B has only vertical asymptotes. We can eliminate that since we know we have horizontal asymptotes as well. Option C has the vertical asymptotes at X equals negative four and X equals four. A horizontal A tote at Y equals negative four. And the domain that we found, OK, those three intervals that we found. And so C is gonna be the correct answer. Option D had the horizontal and vertical ascent totes mixed up and the domain would have only had two intervals. OK? And so that was not the complete domain. And so what we have here is option three. 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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