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]