Asymptotes curve (1-2x)/(3x+5)

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Discussion Overview

The discussion revolves around finding the vertical and horizontal asymptotes of the function (1-2x)/(3x+5). Participants explore strategies for determining these asymptotes, including limits and algebraic manipulation, while addressing the implications of these asymptotes in relation to the function and its inverse.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant seeks assistance in finding the vertical and horizontal asymptotes of the function.
  • Another participant suggests checking the denominator for vertical asymptotes and taking the limit as x approaches infinity for horizontal asymptotes.
  • A participant expresses confusion about the complexity of the answer and proposes an algebraic manipulation to find values that y cannot take.
  • There is a discussion about the relationship between horizontal asymptotes of a function and vertical asymptotes of its inverse, with a participant providing a general form for this relationship.
  • One participant notes that when the degrees of the numerator and denominator are the same, the horizontal asymptote can be found by taking the ratio of the leading coefficients.
  • Another participant elaborates on the method for finding horizontal asymptotes by dividing by the highest power of x and taking limits, emphasizing the definition of asymptotes.

Areas of Agreement / Disagreement

Participants present various methods and interpretations for finding asymptotes, with no clear consensus on the best approach or final answers. Some participants agree on certain methods, while others express uncertainty or propose alternative views.

Contextual Notes

Participants discuss the implications of their findings and the algebraic manipulations involved, but there are unresolved mathematical steps and assumptions regarding the function's behavior at infinity and the nature of its inverse.

JohnnyPhysics
I the curve (1-2x)/(3x+5). I have been asked to find the verticle and horizontal asymptotes. Can anyone help me with a strategy?
 
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Here are some hints.

Vertical asymptote: check the demominator, see which point doesn't exist.

horizontal asymptote: take limit of f(x) as x tends to infinity. What is the relation between the limit value you find and the horizontal asymptote?
 
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horizontal asymptote: take limit of f(x) as x tends to infinity. What is the relation between the limit value you find and the horizontal asymptote?
I don't know but that answer looks a little too complicated...

y = (1-2x)/(3x+5)
3xy + 5y = 1 - 2x
x = (1 - 5y )/(3y + 2)

Now, find the value y can't have...

but beware, I have been wrong before (intentionally of course! )
 
Originally posted by FZ+
I don't know but that answer looks a little too complicated...

y = (1-2x)/(3x+5)
3xy + 5y = 1 - 2x
x = (1 - 5y )/(3y + 2)

Now, find the value y can't have...

but beware, I have been wrong before (intentionally of course! )

That's interesting. I never thought of it like that. That suggests to me that the horizontal asymptote of a function should be equivalent to the vertical asymptote of its inverse (if the inverse exists). Let's see. That might be too general.

Say y(x)= (Ax^n+B)/(Cx^n+D)
Then y=A/C is its horizontal asymptote.
It's inverse:
yCx^n+yD=Ax^n+B
x^n(yC-A)=B-yD
x=[(B-yD)/(yC-A)]^(1/n)
Has as its vertical asymptote y=A/C
YEAH!
(For the second graph, x is a function of y - so y=A/C is a vertical line).

Of course, the converse should also be true.
I like that. It shows how arbitrary our placement of the axes and the definition of our variables are.
My algebra skills break down from there. I tried having nonzero coefficients for other powers of x [eg. x^(n-1)] but I'm not sure I can solve for x in that case.
 
In this case the largest degrees of the variables are the same (to the first). So for horizontal asymtotes don't u just take the ratio:

-2x + 1
3x + 5

So the horizontal asymtote is (-2/3)

Right?
 
right
To find horizontal asymptotes (when dealing with rational functions of polynomials), divide both top and bottom of the fraction by the highest power of x and take the limit as x->[oo]
Essentially, all terms which have an x that is less than the highest power will tend to zero, and that is a shortcut that most people use.

For example, if the highest power of x is in the denominator, all the terms in the numerator will tend to zero and that asymptote is y=0.

btw, an asymptote is a line (or some y-value) that the graph approaches, so it should be written as y=k, rather than just k, although I'm sure anyone would know what you mean if you said the horizontal asymptote is -2/3
 
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