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f(x)=6/x+5

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2 Answers

Hi Lauren!
 
If your equation is f(x) = __6__, I agree with Philip. In a quotient, a vertical asymptote always occurs where
                                        x + 5
the denominator = 0, so here that would be x = -5. The horizontal asymptote is at y = 0, thought which class you're in has an effect on your reasoning. If you're in Calculus, you can look at the limit as x approaches infinity, as Philip said. If not, there are three ways to approach it. First, you can try really high values of x (like x = 100, x = 1000, etc) and you can see that the y values are getting tiny. Second, you can look at a graph of the function and you'll see that the graph gets closer and closer to the y axis (x = 0). Third, you can consider the "behavior" of the graph. This guy will act like 1/x. If you know your basic graphs, this can help you as well.
 
Oblique asymptotes occur when the highest power of x on the top is exactly one more than the highest power of x on the bottom. That's not the case here, so there are no oblique asymptotes.
 
HOWEVER....
 
If your equation is as you wrote it, f(x) = _6_  + 5, you have a different beast. Before we can determine
                                                                  x
anything from this guy, we have to make it look like a complete quotient, which is what we're used to. To do that, we need a common denominator, which in this case is x. We have to multiply the 5 by x/x, then add the numerators:
 
f(x) = _6_  + _5x_ = _6 + 5x_
            x           x            x
 
This changes everything. A vertical asymptote again happens whenever the denominator = 0, so in this case, that's at x = 0. For the horizontal asymptote, we can use any of our methods.
 
  1. The limit as x = 0 of f(x) is 5.
  2. Trying high values of x will show that the y values are getting closer to 5.
  3. Looking at the graph will show that there is an invisible barrier that the equation doesn't cross...a horizontal line at y = 5.
  4. Looking at the behavior, we see that the highest power on the top and bottom are equal. in this case, we look at the coefficients of those terms that have the highest power of x. Here, that would be the 5 on top and a 1 on the bottom. This simplifies to 5... again showing a horizontal asymptote at y = 5.
 
Anyway you cut it, there is a horizontal asymptote at y = 5.
 
As before, for an oblique asymptote, the power on top has to be exactly one more than the power on bottom. That's not the case, so no oblique asymptote.

Comments

Hmm. The formatting turned out a little weird, so let me try again...
 
The first case is f(x) = __6__
                                   x + 5.
 
The second case is f(x) = _6_ + 5.
                                         x
Let's see if that looks better..... sorry for that!
Darlene - there are no rules, but I consider it a matter of professional respect for tutors not to over-post another tutor's answer.  If the question is answered, please leave it be and go answer an unanswered question.  We all like to see out smiling face on the answers page, and no one likes to be over-posted.
Philip, I was adding the solution to the question typed, without the parentheses. If the equation is as you assumed, with the parentheses, I agreed with you. I was also providing alternates to the limit approach you used, since Lauren may not have studied limits yet. Personally, I don't care about seeing my face; I care about helping students understand. Sorry you were offended; that was not at all my intent.
Darlene,
Philip's answer was the correct way. It's hard to type equations on here. But maybe you can help with my last part. I need to know what the asymptotes are telling us? This part confuses me.
Thank you.
Asymptotes are places where the graph (and the values) get closer and closer to a certain number but never actually reach it. The simplest case is f(x) = 1/x. Think of what happens as you increase x-values...
 
f(2) = 1/2 = .5
f(4) = 1/4 = .25
f(100) = 1/100 = .01
f(10000) = 1/10000 = .0001
 
Notice that the values are getting close to 0. However, the value of the equation can never BE zero, because there's nothing you can divide 1 by that will give you zero as a result. That means there is a horizontal asymptote at y = 0, because the result can't BE zero but it gets closer and closer to it.
 
Vertical asymptotes are similar, but instead of looking at the results, you look at the equation itself. In this case, we can't divide anything by 0, so x = 0. Let's look at values close to 0.
 
f(.01) = 1/.01 = 100
f(.001) = 1/.001 = 1000
f(.00001) = 1/.00001 = 10,000
 
Notice that when you use x values that are closer to zero, your results get super big. And they will keep getting bigger as you get closer to zero. When you have a case where the results around a certain x-value keep going up and up and up (or down and down and down) without actually being able to put the number in the equation, you have a vertical asymptote.
 
Do you know what asymptotes look like on a graph?
f(x) = 6/(x+5)
 
There will be a vertical asymptote whenever the denominator = 0, which occurs at x = -5.
 
f(x) --> 0 as x --> ±∞, so the horizontal asymptote is the x-axis (y=0)
 
For x>-5, f(x) is positive.  For x<-5, f(x) is negative

Comments

This depends on what your teacher expects for meanings. For me as a teacher, I would expect to see the following.
 
As x approaches 5 from the left, y gets larger and larger, extending to infinity.
As x approaches 5 from the right, y gets smaller and smaller, extending to negative infinity.
As x gets larger, y approaches zero.
As x gets smaller, y approaches zero.