Help with inverse functions

Questions
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1

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).

Open in Wolfram Cloud ↗

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

Do I write the query correctly?

2

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

3

Screenshotfrom2024-12-1615-55-39.png
Screenshotfrom2024-12-1615-55-39.png797×635 39.4 KB

4

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]
5

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?

6

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))) &]]