Message Boards Message Boards

0
|
252 Views
|
5 Replies
|
0 Total Likes
View groups...
Share
Share this post:

Help with inverse functions

Posted 1 month ago

Hello!

I am pretty new to Mathematica. I actually started to learn it for solving a specific problem: finding the inverse of the following function:

$f(x) = \sum_{i=1}^n \frac{b_i}{1 + e^{c_i(x-a_i)}}$ for a specific $n \in \mathbb{N}$

The $a_i, b_i, c_i \in \mathbb{R}^+$ are constants. I can solve the trivial case of $n = 1$ by hand and it agrees with the result provided by WolframAlpha and Mathematica. The case of $n = 2$, written explicitly (without the sum notation), can be solved by WolframAlpha, but on Mathematica I do not get an explicit formulation (see below).

Could you please help me to get an explicit solution for the case $n = 2$ and higher?

Do I write the query correctly?

POSTED BY: Michele Galasso
5 Replies

It looks like it has interpreted t (x - r) not as a product but as a function t[x - r], and it has inverted with respect to r, not x:

Solve[b/(1 + E^(c (-a1 + x))) + s/(1 + E^t[ -r + x]) == y, r]
POSTED BY: Gianluca Gorni

Thank you Gianluca, you are right! So the question is, can Mathematica give me the explicit form of the solution for n = 2 and higher? If not, why?

POSTED BY: Michele Galasso

I very much doubt that it can be done it in terms of all those parameters. You stand a chance for specific numerical values of the parameters, for example

With[{b1 = 1, b2 = 1, c1 = 1, c2 = 3, a1 = 0, a2 = -1},
 InverseFunction[b1/(1 + E^(c1 (# - a1))) +
    b2/(1 + E^(c2 (# - a2))) &]]
POSTED BY: Gianluca Gorni

enter image description here

POSTED BY: Michele Galasso

Can Wolfram Alpha give you the full solution for n=2 in terms of all six symbolic parameters? What is it like?

POSTED BY: Gianluca Gorni
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract