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) = √(x2-16)f(x) = \(\sqrt{x^2-16}\)
Answer» f(x) = √(x²-16) We know, The square of a real number is never negative. Clearly, f(x) takes real values only when x² – 16 ≥ 0 ⇒ x² – 4² ≥ 0 ⇒ (x + 4)(x – 4) ≥ 0 ⇒ x ≤ –4 or x ≥ 4 ∴ x ∈ (–∞, –4] ∪ [4, ∞) Thus, Domain of f = (–∞, –4] ∪ [4, ∞) When x ∈ (–∞, –4] ∪ [4, ∞), We have, x² – 16 ≥ 0 Hence, \(\sqrt{x^2-16}\) ≥ 0 ⇒ f(x) ≥ 0 ∴ f(x) ∈ [0, ∞) Thus, Range of f = [0, ∞)

Find the domain and range of each of the following real valued functions : f(x) = √(x2-16)f(x) = \(\sqrt{x^2-16}\)

Answer»

f(x) = √(x²-16)

We know,

The square of a real number is never negative.

Clearly,

f(x) takes real values only when x² – 16 ≥ 0

⇒ x² – 4² ≥ 0

⇒ (x + 4)(x – 4) ≥ 0

⇒ x ≤ –4 or x ≥ 4

∴ x ∈ (–∞, –4] ∪ [4, ∞)

Thus,

Domain of f = (–∞, –4] ∪ [4, ∞)

When x ∈ (–∞, –4] ∪ [4, ∞),

We have,

x² – 16 ≥ 0

Hence,

\(\sqrt{x^2-16}\) ≥ 0

⇒ f(x) ≥ 0

∴ f(x) ∈ [0, ∞)

Thus,

Range of f = [0, ∞)