Complete the following steps for the given functions.

Question

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

$f (x) = \frac{3 x^{2} - 2 x + 5}{3 x + 4}$

Accepted Answer

Welcome back everyone. In this problem, we want to identify the vertical asymptotes. If any of the function H of X equals five X squared plus three, X minus seven, divided by two X plus six A says it's X equals three, BX equals negative three and X equals three CX equals 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. And 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 for us to identify the vertical asymptotes, first, let's find the value that makes our denominator zero. Let's check if it makes our numerator zero. And if it doesn't, then let's verify the behavior of the function as X approaches that value from both the left and the right sides. OK. So first, let's start with our denominator. If we let two X plus six be equal to zero, then two X is going to be equal to negative six. And thus X is going to be negative three. Now, at this point, does it make the numerator equal to zero? Well, if we substitute it into our function, so that is we find H of negative three, it's going to be sorry, not of negative three, substitute X equals negative three in the numerator. OK. So in the numerator, when X equals negative three, then a numerator is going to be five multiplied by negative three squared plus three, multiplied by negative three minus seven. That's going to be equal to five, multiplied by nine minus nine minus seven. OK. That's 45 minus 16 which equals 29. So the numerator, OK, the numerator is not equal to zero. Now that we know that's the case, then let's verify the behavior of the function as X approaches negative three from both the left and right sides. OK? No, starting with the left hand side as X approaches. Let's write that here as X approaches negative three from the left. OK. OK. We know the denominator is going to be approaching zero from the left. Since the numerator is positive, that means the function will approach negative infinity. In other words, the limit as X approaches negative three of five, X squared plus three, X minus seven divided by two X plus six is going to be equal to negative infinity. Now, what else do we know? Well, we know that as X approaches negative three from the right side, OK, then the denominator, denominator let make sure I'm spelling that properly here is going to approach zero from the right. Since the numerator is positive, that means the function will approach infinity. In other words, the limit as X approaches negative three from the right. OK. And I should have put it from the left. In our previous example of function, this is going to be equal to infinity. So since it does approach infinity and negative infinity from both the left and right sides, this tells us then, eh that the vertical as symp to the vertical as symp to OK. Vertical as sim to it of air traffic, X is X equals negative three. If we look back at our answer, choices that the C 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).

$f (x) = \frac{3 x^{2} - 2 x + 5}{3 x + 4}$

Key Concepts

Vertical Asymptotes

Rational Functions

Polynomial Functions

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