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

Everyone in this problem for the graph of the rational function given below, we're asked to identify any vertical, horizontal or oblique ACIP totes and to write the domain of the function. So given our graph of a rational function drawn in red and we're giving some dashed lines that are gonna help guide us in this problem looking at those ASOS. Now we're gonna come back to the graph in just a minute to work through the problem. But first, let's have a look at the answer choices, we have four answer choices options A through D K and they differ in which asom totes they have, whether they have a horizontal, a vertical or both and they differ in the domain as well. Now going back to our graph to get started with this problem, then let's start with our vertical asom jos So for our vertical asymptotes, when we're looking at a graph, we wanna look for a vertical line that this function is approaching positive and negative infinity at OK. So we want some vertical line whereas the function or as the X values get really close to that line, the function is shooting up to positive infinity or shooting down to negative infinity. And we can see that this dashed blue line is exactly that as our function gets closer and closer to that line and the X values, it's going up to positive infinity from the right hand side and down to negative infinity from the left hand side. And that function never actually touches that line. And so our vertical ascent tode is going to be that line And the equation of that line is X is equal to seven. OK. It's one particular X value and it happens to be seven. So we have a vertical ascent and X is equal to seven. If we compare this to our answer choices right now, we can already eliminate option A K option A only has a horizontal asom to it. We know that we have a vertical asom to. So we'll eliminate option A and we'll go back to our graph to work out the horizontal as choke. Now, now, for a horizontal Aceto, we want a horizontal line where as the X values go off to positive or negative infinity, this function is going to approach that horizontal line. Now, we can see on the right hand side of our graph as X goes bigger and bigger and bigger. This function is approaching the green dotted line. And the same thing on the left as we go to X equals negative infinity, our function is approaching that same green dotted line. This time from below instead of from above. And so our horizontal axon to is gonna be that green dotted line and that's located at Y is equal to six. We have our horizontal ascent tode at Y equals six. And if we have a horizontal ascent tote, that means that we have no oblique Asymptote. OK. So we don't need to worry about that to have a horizontal aceto where Y is equal to some number that is not zero. The degree of the numerator must be equal to the degree of the denominator of our function. For an oblique aceto, we would need the degree of the numerator to be one more than the degree of the denominator. So we can't satisfy both of those conditions at the same time, which means that having this horizontal Asymptote tells us we have no oblique Asymptote. And again, we can look at our answer choices and we can see that we can eliminate option B now because option B only has a vertical aceto, but we found both a vertical and a horizontal Aceto. So now, option C and D have the same asymptotes that we found and they differ in the domain. So what we need to do is find the domain of this function and the domain of this function is gonna be related to that vertical. As to I recall that the domain is gonna be all possible X values that this function can take. We know what the vertical ascent to that this function approaches positive and negative infinity. OK. So the function actually does not have a defined value exactly at that asks them to and remember that the vertical ascent when you're looking at a function is where the denominator of the function equals zero. OK. So that makes sense if the denominator is zero, we're dividing by zero and that function value will be undefined. And so the domain of this function is gonna be all X values except for X equals seven. We can write that as the interval from negative infinity up to seven union with the interval from seven to infinity where we use round brackets to indicate that we are not including those end points. OK? So seven is not included but everything else is. So one final time looking at our answer choices, we can now narrow it down to the correct answer is option D we found a vertical ascent tode at X equals seven. A horizontal ascent tode at Y equals six and the domain is gonna be all real numbers except for seven. 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 Interpretation

Watch next