Range of rational functions in artificially defined domains

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Consider a rational function (Quadratic/Quadratic form in particular) for which the domain has been artificially defined.In normal cases,when the domain is only a subject to standard restrictions according to the definition of functions we assume the rational function equal to y and form a quadratic in x and then put the discriminant greater than or equal to 0.But how to proceed in such cases?(besides maxima and minima)