How to solve a set of equations where the unknowns are a function and some parameters?

Question

I’d like to know how to solve something like this: \begin{align} f(f(x_2)-f(x_1)) & = 27.5\\ f(f(x_3)-f(x_1)) & = 21.6\\ f(f(x_4)-f(x_1)) & = 15.1\\ f(f(x_5)-f(x_1)) & = 10.2\\ f(f(x_6)-f(x_1)) & = \ \ 8.8\\ f(f(x_7)-f(x_1)) & = \ \ 8.4\\ f(f(x_8)-f(x_1)) & = \ \ 7.8\\ f(f(x_9)-f(x_1)) & = \ \ 6.4\\ f(f(x_4)-f(x_3)) & = 46.5\\ f(f(x_5)-f(x_3)) & = 17.6\\ f(f(x_5)-f(x_4)) & = 28\\ f(f(x_7)-f(x_4)) & = 20.1\\ f(f(x_9)-f(x_7)) & = 22 \end{align} where I’m looking for $f(x)$ and $x_1$, $x_2$, $\ldots$, $x_9$. It is given that $0<x_1<x_2<\ldots< x_9$ and that $f(x)$ is a decreasing function which we have reasons to believe is linear. (Also, there’s a margin of error of about $\pm 0.2$.)

Answer 1

I see that it is impossible to obtain the exact solution for this question, since it needs a numerical solution. I am interested because I see it may be a real world calculation, based on the numbers given.

I will propose two methods.

(1) Since there are 13 equations and 9 $x_i$, we can propose a $f(x)$ with 4 parameters. For example, let: $$f(x) = ax^3+bx^2+cx+d$$ This system of equations would be very complex, and need a numerical way to find the solution.

(2) Assume that: $$f(x) = ax+b$$ and then use least square method to find the best fit of the line.