I need to fit a function to this particular set of data points, but I can't find anything with this general form. I have found a great fit for the inverse of this graph using the form
$$x(y) = \frac{a}{b\exp{(cy-y_c)}+d}+e$$
with the values $a=25.0989,\, b=0.2459,\, c=3.1954,\, d=0.0961,\, e=2.8681,\, y_c=-2.1852$, but inverting this function gives me
$$y(x) = \left[\ln{\left(\frac{a}{x-e}-d\right)} - \ln{(b)} + y_c \right]/c$$
at, say, x = 0 or x = 265 the logarithm is undefined. How can I obtain a function valid over the entire range [0, 265]?
Data points: $$x = [0, 5, 17, 27, 38, 74, 137, 161, 184, 209, 239, 253, 265]$$ $$y = [2.8808, 0.4771, 0, -0.2218, -0.3979, -0.6990, -1.0000, -1.0969, -1.2218, -1.3979, -1.6990, -2.0000, -3.0000]$$
Edit: included data values;
Edit 2: rewrote fit equation in terms of y for coherence; added inverse function to better demonstrate the issue; added calculated coefficients;
Edit 3: change 'e' to 'exp' for clarity.
Even though I didn't find a better analytical solution, most of my data points ended up falling within the range of the logarithm function, so I just defined a piecewise function instead, setting the values I wanted for outlier points. An ugly solution, but works.