Creating a Function with Changing Variable Values for Precise Results

  • Context: High School 
  • Thread starter Thread starter hynpf
  • Start date Start date
  • Tags Tags
    Function
Click For Summary

Discussion Overview

The discussion revolves around the creation of a mathematical function, f(x), that maps positive real numbers to a range between 0 and 1, with specific values provided for certain inputs. Participants explore potential forms of the function and methods for deriving it based on the provided characteristics.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant suggests that f(x) appears to be roughly exponential, proposing a form of f(x) = e^{-ax} where a is a real number.
  • Another participant recommends using a spreadsheet to graph x against ln(f(x)) and perform a linear regression to determine the value of a.
  • There are suggestions for using different functional forms and maximizing R-squared values in trendline analysis to find the best fit.
  • A later reply proposes considering an inverse tangent curve centered at x=500, suggesting it might fit the desired characteristics.
  • Some participants point out a misunderstanding regarding the notation used to describe the limits of the function, specifically the notation for the points approaching 0 and 1.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate form of the function and methods for analysis. There is no consensus on a specific function or approach, and the discussion remains open-ended.

Contextual Notes

Participants note that the function does not need to fit all sample values precisely, and there are various interpretations of how to approach the problem mathematically. The discussion includes some confusion regarding notation and the implications of limits.

hynpf
Messages
2
Reaction score
0
Hi,

I'm trying to come up with a function, but I have no idea how.

Here's my idea of f(x):


f(5000) = 0.07
f(1000) = 0.25
f(900) = 0.28
f(700) = 0.3
f(500) = 0.5
f(200) = 0.8
f(100) = 0.9
f(50) = 0.97
f(1) = 0.99
f(0.5) = 0.999

where 0 < f(x) < 1

as f(x) approaches 0/1, the change becomes less and less noticeable, and becomes closer and closer to 0/1 but never become 0/1.

Thanks!
 
Last edited:
Physics news on Phys.org
Thanks for the input.

I edited my post right before you posted, so here's a summary of characteristics:

f(x) can only be a positive real number between 0 and 1 exclusive.
x can be between 0 exclusive and infinite, but I want f(500) to be around 0.5, and I don't really care what f(x) will be after x = 500.

I don't need a precise function that fits all my sample values; those are just to illustrate what kind of curve the function takes. As long as it satisfies these conditions, I can take any function.

I'm no mathmatician, so I hope I explained this well..
 
Looks very roughly exponential, of the form [tex]f(x)=e^{-ax}[/tex], a is some real number. You could graph x against [itex]\ln f(x)[/tex] in a spreadsheet like Excel or OpenOffice, and do a linear regression to find <i>a</i>.[/itex]
 
Last edited by a moderator:
rachmaninoff said:
You could graph x against [itex]\ln f(x)[/tex] in a spreadsheet like Excel or OpenOffice, and do a linear regression to find <i>a</i>.[/itex]
[itex]If you put the data into Excel and graph the points (use the wizard); you can right-click on any of the data points and select Insert Trendline. In the Trendline dialog box, first click on its Type tab and select exponential or power, then go to Options tab and check "Display equation" and "Display R-squared" boxes. You can copy and paste the same graph over and over, then use different functional forms in each. You might want to select the functional form that maximizes the R-squared.[/itex]
 
0/1 would be 0.
 
matt grime said:
0/1 would be 0.

He was using incorrect notation to describe the point [tex]\left( 0, 1 \right) \in \mathbb{R}^2[/tex].
 
I would suggest perhaps an inverse tan curve, centred at x=500 and offset by an appropriate amount.

Claude.
 

Similar threads

High School How do I invert this exponential function?
  • · Replies 3 ·
Replies
3
Views
4K
High School Let f_n denote an element in a sequence of functions that converges
  • · Replies 7 ·
Replies
7
Views
3K
High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
8K
High School What does the derivative of a function at a point describe?
  • · Replies 36 ·
2
Replies
36
Views
5K
Undergrad Transformation of Functions: How Do Domain and Range Change?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Why can a smooth function be described with fewer terms in a Fourier series?
  • · Replies 3 ·
Replies
3
Views
3K
High School Trying to Understand George Boole's "Laws of Thought"
  • · Replies 5 ·
Replies
5
Views
2K
High School Having trouble deriving the volume of an elliptical ring toroid
  • · Replies 4 ·
Replies
4
Views
5K
Undergrad In which sense(s) do square integrable functions go to zero at infinity?
  • · Replies 2 ·
Replies
2
Views
3K
High School Inner product of functions of continuous variable
  • · Replies 7 ·
Replies
7
Views
3K