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

asked 2021-06-08
What is the domain of \(\displaystyle{f{{\left({x}\right)}}}={5}\frac{{x}}{{{3}-{\left(\sqrt{{x}}-{2}\right)}}}\)?

Answers (1)


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.
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 \(\displaystyle{x}≥{2}\).
Putting these two conditions together we get that the domain is \(\displaystyle{\left[{2},∞\right)}\) without the number 11. Hence
Domain \(\displaystyle{\left({f}\right)}={\left[{2},{11}\right)}∪{\left({11},∞\right)}\)

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