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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.
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.
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.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...?
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.
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.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
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.
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.
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?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
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?
A function with two horizontal asymptotes and three vertical asymptotes is a mathematical function that has two lines (horizontal asymptotes) that the function approaches as the input values become infinitely large or small, and three lines (vertical asymptotes) that the function cannot cross or touch.
The equations of the horizontal asymptotes can be determined by taking the limit of the function as the input values approach positive and negative infinity. The equations of the vertical asymptotes can be found by setting the denominator of the function equal to zero and solving for the input values that make the denominator zero.
Yes, a function can have more than two horizontal asymptotes and three vertical asymptotes. The number of asymptotes depends on the behavior of the function as the input values approach infinity or approach values that make the denominator of the function equal to zero.
The multiple asymptotes in a function can provide information about the behavior of the function as the input values become extremely large or approach values that make the denominator equal to zero. This can help in understanding the overall shape and behavior of the function.
To graph a function with two horizontal asymptotes and three vertical asymptotes, you can first determine the equations of the asymptotes. Then, plot these lines on the graph as dashed lines. Next, plot points on the graph that satisfy the function. As the input values approach infinity or approach values that make the denominator equal to zero, the points should approach the asymptotes. Finally, connect the points to create a smooth curve, and label the graph accordingly.