let be the equation $ y-f(x)=0 $ the idea is to get $ s=g(y) $ that is x as a function of 'y'
can this be made by a root finding algorithm ?? i mean you treat $ y $ as a numerical free parameter and find the roots of $ y-f(x)=0 $ in general these roots will depend on 'y' so we can represent every solution of $ y-f(x) $ for different parameters of 'y'
for example in Newton's method
$ x_{n+1} (y)=x_{n} (y)- \frac{y-f(x)}{-f'(x)} $