Assuming that you are looking at a real function of a real variable you can determine the allowed domain as those values of t that produce a real result for g. In this case you need otherwise you are trying to get the square root of a negative number. Factorising with solutions and . So the allowed domain is and . The corresponding range is the values that g ranges over given this domain which can be seen to start from 0 (if or and extend to

This changes if you allow g to be a complex function of a real variable, or a complex function of a complex variable.

The domain of a function is also often specified as a subset of the allowed domain, so you might have a function like g restriced by definition to a range .

We have The domain of is:

• Let we consider the first inequality:
  To make the explanation clearer let we consider to negation: Because is not solvable( can not be negative)
  The solution is , because we considered the negation, so we must negate it again what result Let denotes the first solution set, so
• Now let consider the second inequality:
  This inequality is already solved. In analogue to the first case let denotes the second solution set, so
• Now let consider the last inequality:
  
  The whole solution

As you say, the domain of a function is part of its definition. The "definitions" of $f$ and $g$, in the context of the question, are not definitions at all, but merely equations. I would reformulate the question as: what is the largest domain in $\mathbb{R}$ for which this equation defines a function?