How can I determine the roots of (x)*(1/9)^(1/9)^x - 1 = y using DESMOS?

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Discussion Overview

The discussion revolves around finding the roots of the equation (x)*(1/9)^(1/9)^x - 1 = y using the DESMOS graphing calculator. Participants explore various methods for identifying roots, including numerical approximation and graphical analysis.

Discussion Character

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

Main Points Raised

  • One participant asks how to find the roots of the equation.
  • Another suggests trying specific values like 0, 1, and 2, and encourages plotting the function to understand its behavior.
  • Some participants propose that if analytic solutions are not available, plotting the function and calculating approximations for the roots may suffice.
  • There is a mention of the Lambert W function as a potential non-graphical method to solve the equation.
  • Multiple participants note that the DESMOS graphing calculator shows two zeros within a specified range, but the exact values are left unspecified.
  • One participant expresses difficulty reproducing results from DESMOS, noting discrepancies in the roots obtained during different attempts.
  • There is a discussion about interpreting the expression correctly, particularly regarding the order of operations in exponentiation.
  • Another participant highlights the quirks of the DESMOS input editor that may affect how expressions are entered and interpreted.
  • Confusion arises over earlier misinterpretations of the equation, leading to different results based on how the factors are understood.

Areas of Agreement / Disagreement

Participants express varying opinions on the methods for finding roots, with some favoring graphical approaches and others suggesting numerical methods or the Lambert W function. There is no consensus on the exact roots or the best approach to take.

Contextual Notes

Participants note limitations in their ability to reproduce results and the potential for misinterpretation of the equation's structure, particularly regarding exponentiation. The discussion reflects uncertainty about the behavior of the function and the accuracy of the roots identified.

Who May Find This Useful

This discussion may be of interest to those exploring numerical methods for root-finding, users of graphing calculators like DESMOS, and individuals studying the properties of equations involving exponentiation.

laplacianZero
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[(x)*(1/9)^(1/9)^x ] - 1 = y

How do you find the roots?
 
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If you cannot find analytic solutions, things are easier. plot it, calculate approximations for the root, and you are done.
 
Besides plotting to find approximate root or using the Newton raphson method, are there any other ways?
 
laplacianZero said:
Besides plotting to find approximate root or using the Newton raphson method, are there any other ways?
Since x appears as both the base and as an exponent, the only non-graphical, non-numerical alternative is the Lambert W function. See https://en.wikipedia.org/wiki/Lambert_W_function
 
The desmos graph calculator shows two zeros at ? and at ? where ## 0 < x < 10##

The ? are left to the student.
 
jedishrfu said:
The desmos graph calculator shows two zeros at ? and at ? where ## 0 < x < 10##

The ? are left to the student.
The expression does not have a second zero, at least not with the conventional interpretation of a^b^c as a^(b^c).
 
Yes, you're right. I can't reproduce what I typed into the Desmos calculator.

Last night it gave me a curve that looked something like the Lambert W curves and crossed at x=1.411 and x=9.

Something like this:

##y = \left(x\right)\cdot e^{-x}\ -\frac{1}{5}##

However, now when I plot it, I see only x=1.79.

##y = x\cdot \left(\frac{1}{\left(9\right)}\right)^{\left(\frac{1}{\left(9\right)}\right)^x}-1##
 
Last edited:
You get that if you interpret a^b^c as (a^b)^c = a^(b*c).
 
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  • #10
The DESMOS graphing calculator input editor is a little wonky. Its different in that while it completes parentheses when you type the closing parentheses it add another on instead saying okay got that.

To get around this behavior, you have to instead tab out of the closing parentheses to get where you want to be.
 
  • #11
mfb said:
You get that if you interpret a^b^c as (a^b)^c = a^(b*c).
Yes, that may have what happened and I didn't notice. I should have saved the expression that was input then I didn't where I went wrong.

I do know earlier I had thought the x*(1/9) factors were x^(1/9) instead which produced yet a different result.

One nice thing about the DESMOS is that as you type in the expression and it looks correct you can copy and paste it here as its valid Latex.
 

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