Work each problem. Which function has a graph that does not have ...

Question

Work each problem. Which function has a graph that does not have a horizontal asymptote?

A. ƒ(x)=(2x-7)/(x+3) B. ƒ(x)=3x/(x^2-9) 
C. ƒ(x)=(x^2-9)/(x+3) D. ƒ(x)=(x+5)/(x+2)(x-3)

Accepted Answer

Hey, everyone in this problem, we're asked to identify which of the following rational functions has no vertical asymptotes. We're given four answer choices A through D and we're gonna go through each by one to look at whether or not they have a vertical as to. So let's start with A, OK. In A, we have the function F of X is equal to five X minus six divided by X plus 13. Now recall that a vertical ascent is going to occur where the denominator of the function is equal to zero once this function is simplified. So here we have a linear function in the numerator and the denominator, we cannot simplify this function any further. And so the vertical AYO is gonna be where X plus 13, that denominator is equal to zero, which tells us that we have a vertical Asymptote at X equals negative 13. So for option A, we do have a vertical asom tote. So that is not the correct answer. We can cross that one. Now, moving to option B, we have F of X is equal to seven X divided by X squared minus 25. Now, in this case, we can simplify a little bit. Can we can factor that denominator? So we can write F of X is equal to seven X divided by X minus five X plus five. Yes, we factored it but none of those factors canceled. So there really isn't any more simplifying to do. So, our vertical asymptotes are gonna occur where X minus five is equal to zero or where X plus five is equal to zero. Because if either of those factors are equal to zero, then the denominator will be equal to zero. This gives us vertical asymptotes of X equals five and X equals negative five. So for this function again, we have vertical ascent totes. So that's not gonna be the correct answer. So we can cross out option B let's move to option C. For option C, we have the function F of X is equal to X squared minus 36 divided by X plus six. And you could say, OK, we set the denominator equal to zero. We get a vertical Asymptote. But remember you need to simplify first and that's really, really important. So in this case, we can factor the numerator, OK, we have a difference of squares X squared minus 36. We can factor this as X minus six, multiplied by X plus six. All of that is divided by X plus six. And now you can see that that factor of X plus six is actually going to divide up. And so our function F of X, it's gonna be equal to X minus six and we have no vertical as because there is nothing in that denominator to equal to zero. OK. So this is the function we were looking for, this is the function with no vertical ascent to. OK. So c it looks like it's gonna be the right answer. Let's just go to D and double check that we haven't made a mistake. OK. So the, for the function for D, we have F of X is equal to X plus 17, all divided by X plus three, multiplied by X minus nine. Now, this function again is already simplified. It's factored into linear factors. Nothing can divide out or cancel. And so we're looking where the denominator is equal to zero. OK? If either of these factors is equal to zero, the entire denominator will be equal to zero. So we have vertical asom totes, they're gonna occur where X plus three is equal to zero four, where X minus nine is equal to zero, which tells us that the vertical asymptotes are at X equals negative three or X equals nine. OK. So for D, we did have vertical asymptotes. So that is not the correct answer either. And so C is indeed the correct answer. We found that that's the only function that has no vertical asymptotes. Thanks everyone for watching. I hope this video helped see you in the next One.

Work each problem. Which function has a graph that does not have a horizontal asymptote? A. ƒ(x)=(2x-7)/(x+3) B. ƒ(x)=3x/(x^2-9) C. ƒ(x)=(x^2-9)/(x+3) D. ƒ(x)=(x+5)/(x+2)(x-3)

Key Concepts

Horizontal Asymptotes

Rational Functions

Degree of a Polynomial

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