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. The function given is F of X equals negative one divided by X minus four squared. We're given four answer choices A through D. They are all graphs. OK. Option A shows the function is approaching positive infinity. As X gets closer and closer to four. Option B shows that as we approach that vertical Asymptote at X equals four, the function is approaching negative infinity. Option C shows that as we approach the vertical Asymptote X equals four. From the left-hand side, the function goes to positive infinity. And as it approaches the vertical asom to from the right hand side, it goes to negative infinity. And option D is the opposite. As it approaches the vertical ascent from the left hand side, the function goes to negative infinity. And as it approaches from the right hand side, it goes to positive infinity. We're gonna move down and give ourselves some space to work. And let's rewrite the function we were given, we have F of X is equal to negative one divided by X minus four squared. Now, the numerator is always going to be negative. So we don't need to worry about what happens to the numerator depending on what side we're approaching. The function from the numerator is always negative. Mhm What about the denominator? We have an X in the denominator so that maybe it'll depend what side of the vertical until we're at be careful because there's a squared here. OK? So we have something squared which is always gonna be positive. OK. So it doesn't matter what our X value is, it doesn't matter which side of the Asymptote we're on, the denominator will always be positive. Mhm OK. As long as we are not at that vertical as to where it will be zero, then others add X not equal to four. OK. So the numerator is always negative, the denominator is always positive, which means that our function F of X is always gonna be a negative value divided by a positive value which is going to give us a negative. OK. So our function F of X is always negative. So whether it's approaching from the left hand side or the right hand side, OK. The denominator is going to be getting closer and closer to zero as we get X closer and closer to four. OK? Because we aren't gonna be close to four minus four. So the denominator is getting closer and closer to zero, which means that this function is going to infinity, either positive or negative. We found that our function is always negative and so it must be going off to negative infinity on both the left and right hand sides. If we compare this to our answer choices, we see that this corresponds with answer choice B K and B, we have the function staying negative and going off to negative infinity as it approaches that vertical as to. Thanks everyone for watching. I hope this video helped see you in the next one.

Struggling with College Algebra?

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

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