(i) [–3, –2) U [3, 4)

(ii) R – {(0, 1) ∪ {1, 2,......., 12} ∪ (12, 13)}

(iii)

(iv)

(v).

Sol. (i)

Domain

(i) log 1/3 log 4 ([x] 2 – 5) 0 or log 4 ([x] 2 – 5) 1

or [x] 2 9 or x ∈ [–3, 4) .........(i)

(ii) log 4 ([x] 2 – 5) > 0

or [x] 2 – 5 > 1

or x ∈ (– ∞ , –2) ∪ [ 3, ∞ ) .........(ii)

(iii) [x] 2 – 5 > 0

x ∈ (– ∞ , –2) ∪ [ 3, ∞ ) .........(iii)

Now (i) ∩ (ii) ∩ (iii)

x ∈ [–3, 2) ∪ [ 3, 4)

(ii) f(x) =

Case- Ι x > 12

f(x) = ⇒ f(x) =

Now for f(x) to be defined [x] ≠ 12 ⇒ x [12, 13) but x > 12

Case- ΙΙ 1 ≤ x ≤ 12

f(x) = =

⇒ x {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Case- ΙΙΙ

x < 1

f(x) =

f(x) = ⇒ x (0,1)  x < 1

(iii) f(x) =

x + 0.5 > 0, x + 0.5 1 ⇒ x ∈ (–0.5, ∞ ) & x ≠ 0.5 .....

& > 0 or > 0

or x ∈ (– , –3) .....

∩ ∴ Domain of f(x) : x ∈ –

(iv) f(x) = − +

0 ⇒ x [1, 3)

& x ∈ [–1, 1]

& x + 1 > 0 ⇒ x ∈ (–1, )

& 7x + 1 ∈ w

Domain

(v) 3 y = > 0

4x 2 – 1 > x 4 ⇒ (x 2 ) 2 – 4x 2 + 1 < 0

(x 2 – 2) 2 + 1 – 4 < 0

< |x| <

x ∈

x ∈

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