(i) Given f (x) = –|x|
We know |x| is defined as non-negative real number for every x ∈ R.
∴ Domain of f = R
Also, |x| ≥ 0 ∀ x ∈ R
⇒ – |x| ≤ 0 ∀ x ∈ R
∴ Range of f = (– ∞, 0]
(ii) Given f (x) = √9−x2
We know √9−x2 ≥ 0 ⇒ 9 − x2 ≥ 0
⇒ x2 ≤ 9 ⇒ |x| ≤ 3 ⇒ – 3 ≤ x ≤ 3
∴ Domain of f = [– 3, 3]
Let y = f (x) = √9−x2 ⇒ y2 = 9 – x2, y ≥ 0 ...(i)
⇒ x2 = 9 – y2 ⇒ 9 – y2 ≥ 0 (Since x2 ≥ 0) ⇒ y2 ≤ 9 ⇒ |y| ≤ 3 ⇒ – 3 ≤ y ≤ 3
But from (i), y ≥ 0 ⇒ 0 ≤ y ≤ 3.
∴ Range of f = [0, 3].