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)^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 a vertical Asymptote X equals four. We're given the function F of X is equal to one divided by X minus four squared. We're given four answer choices. All four are graphs that have the vertical Asymptote X equal four drawn in red. Option A shows that as we approach the vertical Asymptote from either the left or the right, the function is approaching positive infinity. Option B shows that as we approach the vertical ay to from the left or the right, the function is approaching negative infinity. Option C shows that as we approach the vertical Asymptote 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 our last option option D shows that as we approach that 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 go ahead and write out the function we were given and the asom to we were given as well. We have the function F of X is equal to one divided by X minus four squared and the vertical as until were given is X equals four. Now, if we think about approaching this Asymptote from the left, what happens? Well, if we're to the left of X equals four, that means our X values are going to be less than four that tells us that X minus four is going to be less than zero and it's going to be negative. But our denominator has X minus four squared and so X minus four squared is actually going to be positive. It's gonna be bigger than zero. OK? A negative times a negative gives us a positive anything squared is always going to be positive. So our function F of X, it's going to be equal to a positive value divided by a positive value which gives us a positive value. So as we approach that vertical ay, the denominator is gonna be getting closer and closer to zero because our X values are getting closer and closer to four. So this function is growing and it's going to positive or negative infinity. And we figured out that from the left hand side, our function values are positive. And so this is gonna be going off to positive infinity. Now, from the right hand side, we can do the same thing. In this case, on the right hand side of X equals four our X values are going to be greater than four, which tells us that X minus four is going to be greater than zero. The factor X minus four squared is also going to be greater than zero. And so similarly to the left hand side, our function F of X going to be equal to a positive value divided by a positive value, which gives us a positive value for our function F of X. So from the right hand side, we have the same thing that denominator is getting closer and closer to zero, our function values are positive. And so we're going off the positive infinity. From the right hand side, we compare this to our answer choices. We found that from both the left and the right hand side as we approach this vertical Asymptote X equals four, our function is going off to positive infinity. And so we have option A, that's it for this one. Thanks everyone for watching. I hope this video helped.

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)^2

Key Concepts

Rational Functions

Vertical Asymptotes

Behavior Near Asymptotes

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