avissidep

2021-06-08

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

bahaistag

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 $3-\sqrt{x}-2=0$. So let's find out which x's make this true so that we can exclude them from the domain.
$3=\sqrt{x-2}$
$9=x-2$
$11=x$
So 11 cannot be in the domain.
The number under the radical cannot be negative. That is, we must have $x-2\ge 0$. Hence $x\ge 2$.
Conclusion
Putting these two conditions together we get that the domain is $\left[2,\infty \right)$ without the number 11. Hence
Domain $\left(f\right)=\left[2,11\right)\cup \left(11,\infty \right)$

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