# Help with inverse functions

**URL:** <https://community.wolfram.com/t/help-with-inverse-functions/24880>\
**Category:** Questions\
**Tags:** wolfram-alpha, calculus\
**Created:** [December 16, 2024, 12:02pm UTC](https://community.wolfram.com/t/help-with-inverse-functions/24880 "2024-12-16T12:02:00Z")\
**Posts on this page:** 6\
**Page:** 1

<div class="post-metadata">

**Author:** ![galasmic](https://avatars.discourse-cdn.com/v4/letter/g/35a633/32.png) [@galasmic](https://community.wolfram.com/u/galasmic)\
**Post date:** [December 16, 2024, 12:02pm UTC](https://community.wolfram.com/t/help-with-inverse-functions/24880/1 "2024-12-16T12:02:00Z")

</div>

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 ↗](https://www.wolframcloud.com/obj/de08c106-1f1c-4a23-835e-259641e9e9e4)

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

Do I write the query correctly?

---

<div class="post-metadata">

**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [December 16, 2024, 1:29pm UTC](https://community.wolfram.com/t/help-with-inverse-functions/24880/2 "2024-12-16T13:29:28Z")

</div>

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

---

<div class="post-metadata">

**Author:** ![galasmic](https://avatars.discourse-cdn.com/v4/letter/g/35a633/32.png) [@galasmic](https://community.wolfram.com/u/galasmic)\
**Post date:** [December 16, 2024, 2:57pm UTC](https://community.wolfram.com/t/help-with-inverse-functions/24880/3 "2024-12-16T14:57:11Z")

</div>

![Screenshotfrom2024-12-1615-55-39.png](https://global.discourse-cdn.com/wolfram/original/2X/e/e1a811b2c9d30984e2981decf74ffef788dc93d4.png)

---

<div class="post-metadata">

**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [December 16, 2024, 4:16pm UTC](https://community.wolfram.com/t/help-with-inverse-functions/24880/4 "2024-12-16T16:16:22Z")

</div>

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]

```

---

<div class="post-metadata">

**Author:** ![galasmic](https://avatars.discourse-cdn.com/v4/letter/g/35a633/32.png) [@galasmic](https://community.wolfram.com/u/galasmic)\
**Post date:** [December 16, 2024, 7:25pm UTC](https://community.wolfram.com/t/help-with-inverse-functions/24880/5 "2024-12-16T19:25:09Z")

</div>

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?

---

<div class="post-metadata">

**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [December 16, 2024, 7:58pm UTC](https://community.wolfram.com/t/help-with-inverse-functions/24880/6 "2024-12-16T19:58:01Z")

</div>

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))) &]]

```