If there was a function f(x,y) = z...would the inverse be defined under injection (one-to-one correspondence) only if this condition was held?: Every z corresponds to only one (x,y) pair. If this was the case...then wouldn't that imply that a good number of 2-D surfaces cannot have an inverse mapping? Furthermore...what multivariable linear mapping would have an inverse (its not too hard to imagine multiple different orders of variables for which the same value is designated)? Edit: my question is more or less this: What are the necessary conditions for an inverse for a multivariable mapping? If one-to-one correspondence is one of them, then what is the definition of one-to-one correspondence for multivariable mappings?