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.