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.

Graph of y=f(x) showing vertical, horizontal, and oblique asymptotes.

Accepted Answer

Hey, everyone in this problem for the graph of the rational function given below, we are asked to identify any vertical, horizontal, oral bleak asymptotes. We're also told to write the domain of the function. So we're giving our graph, OK. Our function is drawn in red and we have some dotted lines drawn in blue and green that are gonna help us with these asom totes. And we're given four answer choices, option A through D and they contain different information about the asymptotes. OK? There's a mixture of oblique asymptotes, vertical asymptotes or both as well as the domain of this function. Let's go back up to our graph of our function and take a look. Now, let's start with the vertical asymptotes. OK. So we're looking at vertical asymptotes, we wanna look at any vertical lines where this function approaches positive and negative infinity as X approaches those lines or that line. OK. So if we look at our graph, we can see that on the left hand side here as our function gets closer and closer to X equals negative two. From the left, our function is going down to negative infinity and from the right that function is going up to positive infinity. OK. So that function is never going to actually have a value at X equals negative two. It's approaching positive and negative infinity as we get closer to this vertical line, which means that we're gonna have a vertical ASOM tope there. And that vertical ato is located at X equals negative two. And we can pull that from our graph. Now, if we want to do a horizontal as we wanna look for where we have a horizontal line that the function approaches. Looking at our graph, we don't really see a horizontal asom tote and there's no horizontal line that this graph is approaching. So it doesn't look like we're gonna have a horizontal ASOM to and we're gonna leave that for a minute. An oblique ascent tope an oblique ascent to is going to be a line that is not horizontal, that is not vertical that this function approaches. And if we look, we can see that we have this green dotted line that the function is approaching as it goes to positive and negative infinity. OK. So we have a, an oblique Asymptote and it's gonna be this green line. OK. So our oblique ascent to is the line and we need to write the equation of this line. So let's figure out two points and write this line. So we have a line passing through. I want to choose two points that are gonna be simple to work with. So let's choose the X and Y intercepts. OK. So this line passes through the 0.0, negative two and it also passes through the point to comma zero. Now this first point is our Y intercept. Now recall that if we write our equation for the line as Y equals M X plus B, that B represents our X intercept, OK? Because when we substitute X equals zero, we just get Y is equal to B OK. So we know that B must be equal to negative too. So we have the line Y is equal to M X minus two and we can substitute in our other point. So we're gonna substitute into comma zero. Now, when we do that, we get zero is equal to multiplied by two minus two. Moving this negative two to the other side by adding, we get that two, M is equal to two and dividing by two to isolate for F for M. Yep, we get M is equal to one. All right. So our oblique Asy is therefore going to be Y is equal to X minus two. Now, I wanna come back to the horizontal asci tote for a minute. We said that from our graph, it didn't look like we had a horizontal ascent to. Now we found an oblique Asy tope. OK. If we have an oblique Asom Tobe, that means that we cannot have a horizontal Asymptote. OK. The conditions for the two can never match up. OK. So we have an oblique Asymptote that means that we don't have a horizontal Asymptote. And so what we originally said by looking at the graph is correct. OK. So we've done our as and totes. Now, the question is asking for the domain of the function may recall that the domain is all of the possible X values for this function. Now, from our graph, it looks like almost every X value will be OK. But we have to be careful is that this vertical Asymptote. And when we have this vertical ascent tode at X equals negative two, we can see that this function is approaching positive and negative infinity. So this function is not going to be defined at X equals negative two. And that makes sense. Remember that the vertical Asymptote is where the denominator of a rational function is equal to zero. OK. So if the denominator is equal to zero, that function is going off to positive or negative infinity and we have no value there. So for our domain, it's gonna be everything except for negative two. How we write that while we have the interval from negative infinity all the way up to negative two. And we use round brackets to indicate that we don't include the end points. OK? We don't want to include negative two and that's gonna be a union with the interval from negative two up to infinity again round brackets. So we're taking everything from negative infinity to infinity except exactly at negative two. All right. So we found a vertical ascent tode at X equals negative two, an oblique Asymptote at Y equals X minus two and a domain from negative infinity to negative two union with negative two to infinity which corresponds with answer choice. D thanks everyone for watching. I hope this video helped see you in the next one.

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

Key Concepts

Asymptotes

Domain of a Function

Graph Behavior at Infinity

Watch next