In Exercises 21–36, find the vertical asymptotes, if any, and the...

Question

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=(x^2+4x−21)/(x+7)

Accepted Answer

consider the given rational function and fine dreadful asuntos and values of X corresponding to holes. Our equation is X squared minus two, X minus 15 over X plus three. Now the check for aspen tow or hold we want to see if there's a discontinuity in this rational function to do that. We're gonna set our denominator equal to zero and solve however. You first want to check if we have common factors in both the numerator and the denominator because that will determine what classification of this continuity we have. So let's go and take a look. You have X squared minus two X minus 15 on top which can be factored Now we know 15 the factors of 15 we want are going to be three and 5 and one minus and minus two. So I'm gonna say you want x minus five X plus three That is over X-plus three. Now us here we have a common factor on top we have expose three. And on the bottom we have expose three. If it was just X plus three in the bottom with no common factors we would have a vertical as in tow because we have common factors on top and bottom we don't have vertical ascent to anymore we have a whole So the equation of our line. If we were to cancel our X-plus threes, this actually X -5. However, since we cancel our x minus three, X plus threes, we wanna keep that in mind. We have a discontinuity at X plus three based on the original function. So I'm all right. This kind of nudity at X plus three. If we were to solve this for X, let's say equal to zero, We will say x equals -3. Sorry, equation is X minus five with a discontinuity at X equals negative three. Since we have common factors, this is actually a whole and not a vertical ascent. Oh, We have a whole x equals negative three and we have no vertical as because of our common factors, Which means the answer to our problem is answer. See no vertical ascent and a whole at X -3. Okay. I hope to help you solve the problem. Thanks for watching. Goodbye.

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In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=(x^2+4x−21)/(x+7)