Question

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

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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)

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 \(\displaystyle{\left({f}\right)}={\left[{2},{11}\right)}∪{\left({11},∞\right)}\)

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