Answers:

(i)

Now we have,

f(x) is defined for all real numbers x.

Domain (f) = R

If x < 0,

–|x| < 0

f (x) < 0

If x ≥ 0,

–x ≤ 0.

–|x| ≤ 0 ⇒ f (x) ≤ 0

∴ f (x) ≤ 0 or f (x) ∈ (–∞, 0]

Range (f) = (–∞, 0]

(ii)

Square of a real number is not negative.

f(x) takes real values only when 9 – x² ≥ 0

9 ≥ x²

x² ≤ 9

x² – 9 ≤ 0

x² – 3² ≤ 0

(x + 3)(x – 3) ≤ 0

x ≥ –3 and x ≤ 3

∴ x ∈ [–3, 3]

Domain (f) = [–3, 3]

If x ∈ [–3, 3],

0 ≤ 9 – x² ≤ 9

0 ≤ √(9-x²) ≤ 3 ⇒ 0 ≤ f (x) ≤ 3

f(x) ∈ [0, 3]

∴ Range (f) = [0, 3]