The domain of an inverse of a relation is always the domain of the original relation. true or false?

Since an inverse function by definition reverses the coordinates, ie f(x) produces (x,y) the f^-1(x) produces (y,x).

The domain of the inverse function is the domain of the original function. 

So the statement is FALSE unless the domain and range of the original function were the same.  Thus is cannot ALWAYS be true because a function domain and range are not always the same.