Find the domain and range of each of the following real valued functions:
We know the square of a real number is never negative.
Clearly, 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]
Thus, domain of f = [–3, 3]
When x ∈ [–3, 3], we have 0 ≤ 9 – x2 ≤ 9
Hence,
∴ f(x) ∈ [0, 3]
Thus, range of f = [0, 3]