Invertible function y=f(x), x=f^(-1)(y) with two linear segments and smooth transition

[smartscience]

Hi,

I'm looking to find a function y=f(x), invertible to x=f(y) and written in terms of elementary functions and operations, that can represent a straight line Ax+B where x<<T and another straight line Cx+D where x>>T, where T is the x position where the two lines would cross. In the region where x is approximately T, there should be a smooth transition between the two functions. For what it's worth, my intention is to get a functional representation of the force-displacement relationship for a cracked solid, which becomes stiffer when the crack is closed.

Some suitable functions can be found by integrating any sigmoid-like function, or equivalently integrating any peak-shaped function twice. The width of the peak or sigmoid then becomes the width of the transition from one line to the other. Similarly, a sigmoid function could be used as a weighting factor for terms Ax+B and Cx+D, e.g.

y=(Ax+B)(1/(1+exp(x))) + (Cx+D)(1/(1+exp(-x)))

Simpler functions like

y=ln(exp(Ax+B) + exp(Cx+D))

also spring to mind. However, these functions don't appear to be invertible either with my mathematical knowledge as it stands, nor with the computer algebra package Maxima.

Does anyone know any solutions to this kind of problem please? If this is a well-known problem, some terms to search for on would be appreciated. If there's no obvious solution, I'd even welcome just some suggestions for sigmoid functions that are more likely to yield an invertible function when used in this way.

Thanks in anticipation, Joe

 

[Mark44]

Very late response, but in case someone else is interested, this is what I would do. Instead of dividing the interval into two pieces, around x = T, I would divide it into three pieces: $-\infty < A < T < B < \infty$.
The inverse, $f^{-1}$ could be defined in a piecewise fashion on the three separate intervals $(-\infty, A), (A, B), \text{ and } (B, \infty)$. On the left interval you could define the inverse as y = ax + b, and on the right interval as y = cx + d. Notice that a and A etc. are different numbers. A cubic spline could be used to provide a smooth connection between the two outer parts of the graph.