Determine the domain and range of the function?

First off, you can NEVER split the square root of a difference into the difference of two square roots.

You can see this by an example with numbers: does √(25 - 16), which is √9, equal √25 - √16?

Domain is what x can be. Since x is in a square root, and you can't square root a negative, you have to see where 9 - 16x^2 would be less than 0, so first see where it would = 0

9 - 16x^2 = 0

(3 + 4x)(3 - 4x) = 0 so x = -3/4 or x = 3/4. Now if x is between or equal to those, you'd be fine. (You know this by actually picking a number between them, like 0, and plugging in. You get √9 - 0 = √9.) But if x is bigger than 3/4 or smaller than -3/4, 9 - 16x^2 would be negative. So

domain is -3/4 ≤ x ≤ 3/4

Range is what f(x) can be. And since the principal square root of a number is always ≥ 0, range is all numbers ≥ 0