Work each problem. Choices A–D below show the four ways in which ...

Question

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d).

ƒ(x)=1/(x-2)

Accepted Answer

Hey, everyone in this problem, we're asked to identify the graph of the following rational function through the way it approaches the vertical Asymptote X equals four. The function we're given is F of X is equal to one divided by X minus four. We're given four answer choices. All four of them are graphs that show a vertical Asymptote at X equals four graph. A shows that as we approach the vertical Asymptote X equals four from either the left or the right, this function is approaching positive infinity. Option B shows that as we approach this vertical isotope from both the left and the right, the function is approaching negative infinity. Option C shows that as we approach the vertical ascent tope from the left, the function is approaching positive infinity. And as we approach the vertical Asymptote from the right, the function is approaching negative infinity. And option D shows that as we approach the vertical Asymptote from the left, the function is approaching negative infinity. And as we approach the vertical Asymptote from the right, the function is approaching positive infinity. So let's write down the function we have and think about what happens on either side of this vertical asy tope. So we have the function F of X is equal to one divided by X minus four. Now, if we think about approaching the vertical atom to from the left, OK. Our vertical asom tote is at X equals four. So if we're approaching it from the left hand side, that means our X value is going to be less than four numbers on the left hand side on our graph are less in the extraction. All right. So X is less than four. That means that our denominator X minus four is going to be less than zero. OK. It's negative. Now, if our denominator is negative, what happens to our function? Well, our function is going to be a positive value in the numerator divided by a negative value in the denominator, which is going to give us a negative value. So on the left hand side of this Asymptote, just to the left of this Asymptote, we are going to have a negative value for our function as we get closer and closer to the asom tote that denominator gets closer and closer to zero. OK. So we have a negative value. We're getting a denominator, denominator that's getting closer and closer to zero. And so the function is going to be going to negative infinity from the left hand side. If we do the same from the right hand side, what's going to happen. Well, now our X values are going to be greater than four. OK? On a graph, if we're to the right of X equals four, we have X values greater than four. That means that X minus four is going to be greater than zero. OK? It's going to be positive. So if we look at our function F FX and we have one in the numerator which is positive, we're gonna divide it by X minus four, which we know is going to be positive. And so we get a positive value for our function. OK. So just like in the case from the left, the denominator is getting closer and closer to zero. So the function is going off into positive or negative infinity. And we found that on the right hand side, the value of the function is gonna be positive. And so it's going to be going off to positive infinity. So when we look at our answer choices, and we found that from the left, as we approach the vertical Asymptote, X equals four, the function is gonna be approaching negative infinity. And as we approach that vertical asom tope from the right hand side, the function is going to be approaching positive infinity. And so this corresponds with answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.

My Courses

Chemistry

Biology

Math

Physics

Business

Social Sciences

Programming

Product & Marketing

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). ƒ(x)=1/(x-2)