Reverse function numerically

[stn021]

Hello,

I have 2 functions:

y1 = f1( x1,x2 )
y2 = f2( x1,x2 )

also there is the reverse function

x1 = g1( y1,y2 )
x2 = g2( y1,y2 )

in octave syntax:
[ y1,y2 ] = f( [ x1,x2 ] )
[ x1,x2 ] = g( [ y1,y2 ] )

Both functions lead to exactly one distinct pair of results for each
pair of input variables. That means that always

f ( g ( [x1,x2] ) ) == [ x1,x2 ]


My problem is this: I can calculate f(x1,x2) but I cannot calculate g(y1,y2).
Meaning that f( [x1,x2] ) cannot be algebraically reversed.

I am looking for a way to calculate g( [y1,y2] ).
The obvious solution would be some kind of approximation.
(fft looks like a good choice)

So far I could not piece together how to do that.
Could you please give me a hint ?

THX,stn