Domain and Range of sqrt((8x-5)/(x-3))

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sqrt\left(\frac{\left(8x-5\right?}{\left(x-3\right?}\right?

This was an exam question. Is it always that we sketch the rational function and delete the part that goes below the X axis and determine the domain and range?

Does it always work that way?

I saw on the internet that this has complex roots. Is there something that I am missing here?

How do I sketch such graphs? How do I know the domain and range algebraically?

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Also, functions like these sqrt(x^2-9) and sqrt(x^2-6|x|). How do I interpret them and graph them and determine their domain and range algebraically?

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Hello Bruce, could you help me with this please?