Function with two horizontal asymptotes and three vertical asymptotes

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Homework Help Overview

The discussion revolves around finding a function that exhibits two horizontal asymptotes and three vertical asymptotes. Participants explore the characteristics of such functions, particularly in the context of rational functions and their asymptotic behavior.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the requirements for vertical and horizontal asymptotes, questioning how to construct a function that meets both criteria. There are attempts to identify specific functions and their behaviors at infinity.

Discussion Status

Several participants have offered insights into the nature of vertical and horizontal asymptotes, with some suggesting specific forms of functions. There is an ongoing exploration of how to combine different elements to achieve the desired asymptotic behavior, though no consensus has been reached on a specific function.

Contextual Notes

Participants note that vertical asymptotes arise from factors in the denominator of rational functions, while horizontal asymptotes relate to the limits of the function as x approaches infinity. There is also mention of the need for the numerator to behave in a certain way to achieve the required asymptotes.

futoo
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For the life of me I cannot figure out this problem: give an example of a function that has two horizontal asymptotes and three vertical asymptotes. Any help on this topic would be greatly appreciated.
 
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futoo said:
For the life of me I cannot figure out this problem: give an example of a function that has two horizontal asymptotes and three vertical asymptotes. Any help on this topic would be greatly appreciated.

That's a hard one. What's an example of a function that has the two horizontal asymptotes? What about a function that meets the 2nd part of the question? Seems like maybe for the vertical ones, we could use x = zero and +/- infinity... Then how to mix in the horizontal ones...?

We can't give you the answers, but we can try to ask some questions to help you figure it out.
 
berkeman said:
That's a hard one. What's an example of a function that has the two horizontal asymptotes? What about a function that meets the 2nd part of the question? Seems like maybe for the vertical ones, we could use x = zero and +/- infinity... Then how to mix in the horizontal ones...?
A graph could have a vertical asymptote at x = 0, but couldn't have vertical asymptotes at +/- infinity. They would have to occur for some finite numbers. It could have them at, say, x = 1 and x = -1. Vertical asymptotes are determined by factors in the denominator of a rational function that are zero at certain values of x. Horizontal asymptotes arise from the limit of the rational expression being different as x approaches infinity and as x approaches neg. infinity.
 
Mark44 said:
A graph could have a vertical asymptote at x = 0, but couldn't have vertical asymptotes at +/- infinity. They would have to occur for some finite numbers. It could have them at, say, x = 1 and x = -1. Vertical asymptotes are determined by factors in the denominator of a rational function that are zero at certain values of x. Horizontal asymptotes arise from the limit of the rational expression being different as x approaches infinity and as x approaches neg. infinity.


Ah, thanks for the clarification -- my bad. I was just thinking in terms of infinities, but you're right, to be an asymptote, the value would have to go to infinity for some finite argument in the domain of the fuction.
 
since vertical asymptotes arise from 0 being in the denominator, would 1/(x^2-1)(x-3) satisfy having three vertical asymptotes? and wouldn't they be -1, 1, and 3
 
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futoo said:
since vertical asymptotes arise from 0 being in the denominator, would 1/(x^2-1)(x-3) satisfy having three vertical asymptotes? and wouldn't they be -1, 1, and 3
Yes. Now you need to add something in the numerator so that you get two different hor. asymptotes. The function you have has a single hor. asymptote--the line y = 0. All you need to get a different horizontal asymptote is have a third degree polynomial in the numerator whose highest-degree term is x3. There's a bit of a snag, because you want the numerator and denominator to have the same sign for large, positive x, and different signs for very negative x. There's a way around this problem, though.
 
By the way, a "function" does not have to correspond to a single "formula".
 
Mark44 said:
Yes. Now you need to add something in the numerator so that you get two different hor. asymptotes. The function you have has a single hor. asymptote--the line y = 0. All you need to get a different horizontal asymptote is have a third degree polynomial in the numerator whose highest-degree term is x3. There's a bit of a snag, because you want the numerator and denominator to have the same sign for large, positive x, and different signs for very negative x. There's a way around this problem, though.

I also know that tan^-1(x) has two horizontal asymptotes, do I then incorporate x^3 into the numerator of the equation? Also thanks a lot for all your help.
 
Well, tan-1(x) doesn't have anything to do with your problem other than as an example of a function with two hor. asymptotes.

If you have x3 in the numerator, with the denominator you mentioned earlier, you'll get only one hor. asymtote - the line y = 1. Your denominator is negative for very negative x, and positive for very positive x, which is the same for x3.

You need something so that the numerator (only) is positive for large, positive x and for very negative x. HallsOfIvy suggested having two formulas for your function. I'm thinking of something different that involves x3. The basic function I'm thinking of has a V shape.
 
  • #10
Mark44 said:
Well, tan-1(x) doesn't have anything to do with your problem other than as an example of a function with two hor. asymptotes.

If you have x3 in the numerator, with the denominator you mentioned earlier, you'll get only one hor. asymtote - the line y = 1. Your denominator is negative for very negative x, and positive for very positive x, which is the same for x3.

You need something so that the numerator (only) is positive for large, positive x and for very negative x. HallsOfIvy suggested having two formulas for your function. I'm thinking of something different that involves x3. The basic function I'm thinking of has a V shape.

since the numerator has to be positive for large positive x's and negative for negative x's, (x-4)^3/(x^2-1)(x-3) satisfies the requirements that you gave, but when I plug it into my calculator the graph does not show the necessary requirements
 
  • #11
What about a piecewise-defined function? :smile:
 
  • #12
Hint: adding the unit step function to one of the functions already given will give two different horizontal asymptotes and 3 different vertical asymptotes.
 
  • #13
futoo said:
since the numerator has to be positive for large positive x's and negative for negative x's, (x-4)^3/(x^2-1)(x-3) satisfies the requirements that you gave, but when I plug it into my calculator the graph does not show the necessary requirements
You misread what I wrote. The numerator has to be positive for both x << 0 and for x >> 0 (x << 0 means x is very negative). Your function has only one hor. asymptote. If you can keep the numerator positive for large x and very negative x, you'll have two different hor. asymptotes. A simpler function in the numerator is x3. Is there anything you can do to it to make it always >= 0?
 
  • #14
Mark44 said:
You misread what I wrote. The numerator has to be positive for both x << 0 and for x >> 0 (x << 0 means x is very negative). Your function has only one hor. asymptote. If you can keep the numerator positive for large x and very negative x, you'll have two different hor. asymptotes. A simpler function in the numerator is x3. Is there anything you can do to it to make it always >= 0?

an absolute value of a function is always a positive result

I just had this problem in calculus class at PSU

**Edit: realized your question was to teach**
 
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  • #15
Bingo!
Are liquidsnak and futoo the same person?
 
  • #16
No, I was just eager to help out, since I just spent several hours this weekend trying to figure this out.
 

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