Find the domain of each of the following real valued functions of real variable:
We know the square of a real number is never negative.
Clearly, f(x) takes real values only when x2 – 1 ≥ 0
⇒ x2 – 12 ≥ 0
⇒ (x + 1)(x – 1) ≥ 0
⇒ x ≤ –1 or x ≥ 1
∴ x ∈ (–∞, –1] ∪ [1, ∞)
In addition, f(x) is also undefined when x2 – 1 = 0 because denominator will be zero and the result will be indeterminate.
x2 – 1 = 0 ⇒ x = ±1
Hence, x ∈ (–∞, –1] ∪ [1, ∞) – {–1, 1}
∴ x ∈ (–∞, –1) ∪ (1, ∞)
Thus, domain of f = (–∞, –1) ∪ (1, ∞)