How to create an inverted $S$ cruve

Question

I'm looking for a way to create an inverted $S$ curve function. Below is an image of what I want. Left is a regular S curve, on the right is the kind of curve I want. Here are the allowed set of operations that I can use:

Answer 1

Possible with Inverse function where function inversion is possible. The given graph indicates an odd function, straight forward function inversion solution. In polar coordinates $ r(\theta) = r(-\theta).$

In other cases absolute values to be taken and later on modified suitably.

$$ y= f(x)\to x= f(y) $$

$$ or$$

$$ y= f(x)\to y_{inv}= f^{-1}(x_{inv}) $$

In the above we have inverse tanh brown function solution graphed along with starting green tanh function. Similarly the sine function.

Answer 2

I found a function which does what I wanted to do:

$$ f(x)=-\frac{(1-2x)^3}{2}+0.5 $$

Image of the curve

Answer 3

Using complex variables, we can transform the exact $S$-curve you started with very easily. Let us assume the inflection point passes through the origin and let the curve be expressed parametrically as $z(t)$. The first thing we want to do is to flip the curve about the $x$-axis, then rotate it $90^{\circ}$ counterclockwise, the the desired curve becomes

$$w(t)=z^*(t)e^{i\pi/2}=iz^*(t)$$

In Cartesian coordinates then

$$w(t)=i\big(x(t)-iy(t)\big)=y(t)+ix(t)$$

Which means that all you have to do is to switch $x$ and $y$. You can see as much by just flipping your original image about.