(i) For domain (i) [x] > 0 and [x] ≠ 1 so [x] ≥ 2, so x ∈ [2, ∞ )

for range if x ∈ [2, ∞ ), then = 1 so f(x) = cos –1 0 =

Range of f(x) =

(ii) f(x) =

D : 0 < log 2 [x 2 + 4x + 5] ≤ 1

or 1 < [x 2 + 4x + 5] ≤ 2 ⇒ [x 2 + 4x + 5] = 2

or 2 ≤ x 2 + 4x + 5 < 3

D : x ∈ (–2 – , – 3] [–1, –2 + )

R : {0}

(iii) f(x) = sin –1 ⇒ –1 ≤ log 2 < 2 ⇒ ≤ < 4

⇒ x ∈ (– , –1] ∪ [1, ) and R :

(iv) f(x) = log [x – 1] sinx

sin x > 0 ⇒ x ∈ (2n π ,(2n+1) π )

here [x – 1] > 0 & [x – 1] 1 ⇒ x ∈ [3, )

Domain x ∈ [3, π ) .

For range sin x ∈ (0, 1] and [x – 1] ∈ [2, ∞ ) so range ∈ (– ∞ , 0]

(v) f(x) = tan –1 +

Domain : (i) [x] + [–x] ≥ 0 ⇒ x ∈ Ι

(ii) 2 – | x | ≥ 0 ⇒ |x| ≤ 2 ⇒ x ∈ [–2, 2]

(iii) x ≠ 0

For domain (i) ∩ (ii) ∩ (iii)

Domain : {–2, –1, 1, 2}

Range :

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