Find the domain and range of each of the following real valued functio

1.	Find the domain and range of each of the following real valued functions : f(x) = √(9-x2)
Answer» f(x) = √(9-x²) 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, 0 ≤ \(\sqrt{9-x^2}\) ≤ 3 ⇒ 0 ≤ f(x) ≤ 3 ∴ f(x) ∈ [0, 3] Thus, Range of f = [0, 3]

Find the domain and range of each of the following real valued functions : f(x) = √(9-x2)

Answer»

f(x) = √(9-x²)

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,

0 ≤ \(\sqrt{9-x^2}\) ≤ 3

⇒ 0 ≤ f(x) ≤ 3

∴ f(x) ∈ [0, 3]

Thus,

Range of f = [0, 3]