If f (x) = 2-x^1/2 and g (x) + x^2-9, what is the domain of g (x) / f (x) ?

As far a domain, you only care about the denominator and radicals.

f (x) = 2 - √x is the denominator, so it can't be 0.

Also x is under a radical, so x can't be negative.

Set the denominator equal to 0 to find the excluded values.

2 - √x = 0

-√x = - 2

√x = 2

x = 4, so 4 is an excluded value

The domain is all real numbers greater than or equal to 0, except 4.

[0, 4) U (4, ∞)

0 ≤ x 4