(i) (– ∞ ,  n π /2]

(ii) (0, π /2]

(iii) [0, π ]

Sol. (i) f(x) =  n (sin–1x)

Domain sin–1 x > 0 x ∈ (0, 1]

Range 0 < sin–1 x ≤ ⇒ – ∞ <  n (sin–1x) ≤  n

Inequality doesn't change as  n is increasing function

(ii)f(x) = sin–1

 it is obvious is + ve ∀ x ∈ R

for function to be defined ≤ 1 ⇒ ≤ 5x2 + 1  5x2 + 1 > 0 ∀ x ∈ R

squaring both side3x2 + 1 ≤ 25x4 + 10x2 + 1 ⇒ 25x4 + 7x2 ≥ 0

hold for all ∀ x ∈ R. So ≤ 1 ∀ x ∈ R at x → ∞ → 0

So 0 < ≤ 1 ⇒ 0 < sin–1 ≤ π /2

(iii)f(x) = cos–1

form graph , it is clearly visible

that function attain all values b/w [–1, 1]

So Range of cos-1 ∈ [0, π ]

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