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Question # What is the domain of f(x)=5x/(3-(sqrtx-2))?

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ANSWERED What is the domain of $$\displaystyle{f{{\left({x}\right)}}}={5}\frac{{x}}{{{3}-{\left(\sqrt{{x}}-{2}\right)}}}$$? 2021-06-09

That numerator doesn't impose any restrictions, but there are two constraints from the denominator: the whole denominator cannot be 0 and the number inside the radical cannot be negative. Let's look at each in turn.
The denominator cannot be 0 So then we cannot have $$\displaystyle{3}−\sqrt{x}−{2}={0}$$. So let's find out which x's make this true so that we can exclude them from the domain.
$$\displaystyle{3}=\sqrt{{{x}-{2}}}$$
$$9=x-2$$
$$11=x$$
So 11 cannot be in the domain.
The number under the radical cannot be negative. That is, we must have $$\displaystyle{x}−{2}≥{0}$$. Hence $$\displaystyle{x}≥{2}$$.
Conclusion
Putting these two conditions together we get that the domain is $$\displaystyle{\left[{2},∞\right)}$$ without the number 11. Hence
Domain $$\displaystyle{\left({f}\right)}={\left[{2},{11}\right)}∪{\left({11},∞\right)}$$