I have a problem I am hoping to get some help on. I have an equation in two variables (x,y) and one parameter (0<k<1). The equation is =0. My aim is to use this equation to obtain x(y,k) since I need it in another function in the problem. However, the equation is both linear and exponential in x, so I was not able to explicitly solve the equation for x. My next approach was to obtain y(x,k) from the equation, with the idea of then inverting the function. y(x,k) has the form shown below y(x,k) Where f(x) contains all the parts which are exponential in x but do not depend on y or k. This function seems not inversible: as such my idea was to find an approximation of f(x) that is more tractable, and then inverse the function. first, do you think that this approach is matematically correct? of course I will not find the exact solution, but I should still find a good approximation. second, any suggestions on tools for approximating a function? for example, I know f(1)=0 and f(oo)->0.734, so I would need some program/tool that can fit a curve taking these conditions into account.
Thank you all!
Edit: added formula of f(x)
$$f(x) = \frac{(\sqrt{3}-1)(2+\sqrt{3})^x +(1+\sqrt{3})(2-\sqrt{3})^x-2\sqrt{3}}{(2+\sqrt{3})^x+(2-\sqrt{3})^x-2}$$