Inverse function for several variables

[zetafunction]

in one dimension one have that for a function f(x) we can define another function g(x) so f(g(x)=x

my question or problem is the following, if i have a function of three variables f(x,y,z) then i can define another function g(x,y,z) so f(g(x,y,z))=Id

for example for the function f(x,y,z)= xyz what would be its inverse g??

 

[LCKurtz]

in one dimension one have that for a function f(x) we can define another function g(x) so f(g(x)=x

my question or problem is the following, if i have a function of three variables f(x,y,z) then i can define another function g(x,y,z) so f(g(x,y,z))=Id

for example for the function f(x,y,z)= xyz what would be its inverse g??

It doesn't have an inverse because it is not 1-1 onto any point any point in its range. For example if d = abc then d also equals bac, cba etc.