(i) R – {–1, 1} (ii)2n π – ≤ x ≤ 2n π + , n ∈ Ι

(iii) (0, ∞ ) (iv) R

(v) [–2, 0) ∪ (0, 1) (vi)(2, 3]

(vii)(– ∞ , –1] ∪ [0, ∞) (viii) ∪

Sol. (i) f(x) = ⇒ f(x) = . Division by zero is undefined

∴ x ≠ ± 1 ∴ Domain x ∈ R – {1, –1} ⇒ x ∈ (– ∞ , –1) ∪ (–1, 1) ∪ (1, ∞ )

(ii) cos x ∈ [0, 1] ⇒ 2n π – ≤ x ≤ 2n π + , n ∈ Ι

(iii) f(x) = for function to be defined x + |x| > 0 for x > 0, x + |x| = 2x > 0

for x ≤ 0, x + |x| = 0 ∴ Domain is x ∈ (0, ∞ )

(iv) f(x) = e x + sin x . Domain x ∈ R as there is no restriction for exponent of e.

(v) f(x) = +

1 – x > 0 and x + 2 ≥ 0 and 1 – x ≠ 1 ⇒ x ∈ (– ∞ , 1) – {0} and x ≥ – 2 ⇒ x ∈ [–2, 0) ∪ (0, 1)

(vi) Clearly x > 2 and ≥ 0 ⇒ log 2 (x – 2) ≤ 0 ⇒ x – 2 ≤ 1 ⇒ x ≤ 3

(vii) x 2 + x + 1 ≥ 1 ⇒ (– ∞ , –1] ∪ [0, ∞)

(viii) f(x) = ⇒ cos x – ≥ 0 or x ∈ , n ∈ Ι

and 6 + 35x – 6x 2 > 0 or x ∈ ⇒ Domain

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