The logistic equation and its transpose is available : here
The generalized logistic equation is available here. The parameters of this equation can be varied to control its shape and range. $$Y(x) = A+\frac{K - A}{\left(C + Qe^{-B(x - M)}\right)^{1/\nu}}$$
Based on these two sources, I have derived the equation for generalized inverse logistic function as shown here $$X(y) = -\frac 1B\log\left(\frac{\left(\frac {K - A}{y - A}\right)^\nu + C}Q\right) + M$$
However, I cannot control the shape and range as it can be done in the generalized logistic equation. At the end, I want to fix the x and y limits and generate different trends in that region. How can I achieve this?