"What is the domain of f(x)=5x3−(x−2)?" solved and explained

avissidep

Answered question

2021-06-08

What is the domain of

f (x) = 5 \frac{x}{3 - (\sqrt{x} - 2)}

Answer & Explanation

bahaistag

Skilled2021-06-09Added 100 answers

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 $x - 2 \geq 0$ . Hence $x \geq 2$ .
Conclusion
Putting these two conditions together we get that the domain is $[2, \infty)$ without the number 11. Hence
Domain $(f) = [2, 11) \cup (11, \infty)$

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