Solve equation of inverse functions I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and  $f^{-1}_2(x)$, by solving for $x_1$ and $x_2$ two equations
$$
f_1(y_1) - x_1 = 0, \\
f_2(y_2) - x_2 = 0,
$$
for specific realizations of $y_1$ and $y_2$. Since $f_1 \neq f_2$, I get two different solutions for both equations $x_1 \neq x_2$. The problem is that I would like to solve equation on those inverse functions. The goal is to find a single point $x_0$ for
$$
f^{-1}_1(x_0) - f^{-1}_2(x_0) =0
$$
Is this possible? If I plug in the above inverse equations, assuming that $x_1=x_2$, into the main equation I get $f_1(y_1)=f_2(y_2)$ which is a contradiction. Is there a method to solve this problem, even numerically? If not possible, maybe finding $\min |f^{-1}_1(x) - f^{-1}_2(x)|$ would be an option?

This is not always possible. 
Take $f_1(x)=x$ and $f_2(x)=x+1$.
Then $f_1^{-1}(y)=y$ and $f_2^{-1}(y)=y-1$.
Therefore for any $y_0$:
$$f_1^{-1}(y_0)-f_2^{-1}(y_0)=y_0-y_0+1=1\neq 0.$$
