Y = x squared -4x plus 3 is an equation of a function. It is neither called a domain nor a range.
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2 squared plus 2 x 3 = 10, 7 squared plus 7 x 2 = 63, 6 squared plus 6 x 5 = 66,8 squared plus 8 x 4 = 96 so 9 squared plus 9 x 7 = 81 + 63 = 144.
(x^2)^(1/2) equals x, therefore, y = x+4, which has a range and domain of all real numbers. The graph is a straight line, slope of 1, y-intercept of 4. Are you actually saying y = (x^2+4)^(1/2). If so, the range and domain will also be all real numbers because x^2+4 will never result in a negative number.
The function is a simple linear function and so its nature does not limit the domain or range in any way. So the domain and range can be the whole of the real numbers. If the domain is a proper subset of that then the range must be defined accordingly. Similarly, if the range is known then the appropriate domain needs to be defined.
you use A squared plus B squared equals C squared