Complete the following steps for the given functions.

Question

b. Find the vertical asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

Accepted Answer

Welcome back everyone. In this problem, we want to determine the vertical asymptotes. If any of the function H of X equals two X cube plus nine X squared minus five, divided by three X squared plus 27 A says X equals negative nine B says X equals nine C says X equals three and negative three. And D says the function has no vertical Asymptote. Now, what do we know about vertical asymptotes? Well, they occur where the function approaches infinity or negative infinity as the input approaches a certain value, these points often correspond to the denominator of a rational function being zero provided that the numerator does not also become zero at the same point. So if we're going to find a vertical Asymptote, first, we need to figure out the value that makes our denominator zero, then we need to check if it also makes the numerator zero. And if it does not, we can verify the behavior of our function from the left and the right hand side. So first let's try to find the value of our denominator or sorry, the value that will make the denominator of our function zero. OK. Now if we let three X squared plus 27 equal zero, OK, then that means three X squared equals negative 27. If we divide both sides by three X squared equals negative nine and X is going to be equal to the square root of negative nine. But we've run into a problem here. We can't find the square root of a negative number. Or in other words, the square root of a negative number is not real. OK. So that means there are no real solutions. OK. To this equation, since there are no real solutions, that means there is no real value of X that will make the denominator zero. And if that's the case, that means the given function has no vertical Asymptote because remember that the point near far or vertical Asymptote often corresponds to the denominator being zero. Thus D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Complete the following steps for the given functions.

b. Find the vertical asymptotes of $f$ (if any).

$f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)$

Key Concepts

Vertical Asymptotes

Rational Functions

Polynomial Functions

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