The domain of tanh−1(2x−3)4−x2 is

f(x)=Tanh−1(2x−3)4−x2 is definediff −1 0⇒ 2<2x<4 and x2−4<0⇒ 1<x<2 and (x+2)(x−2)<0 ⇒ −2<x<2 x∈(1, 2)

The domain of tanh−1(2x−3)4−x2 is

f(x)=Tanh−1(2x−3)4−x2 is definediff −1 0⇒ 2<2x<4 and x2−4<0⇒ 1<x<2 and (x+2)(x−2)<0 ⇒ −2<x<2 x∈(1, 2)

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$T h e d o m a i n o f \frac{\tanh^{- 1} (2 x - 3)}{\sqrt{4 - x^{2}}} i s$

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$(- 2, 2)$

$[0, 2)$

$[1, 2]$

$(1, 2)$

answer is B.

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Detailed Solution

$\begin{array}{l} f (x) = \frac{{Tanh}^{- 1} (2 x - 3)}{\sqrt{4 - x^{2}}} i s d e f i n e d \\ i f f - 1 < 2 x - 3 < 1 a n d 4 - x^{2} > 0 \\ \Rightarrow 2 < 2 x < 4 and x^{2} - 4 < 0 \\ \Rightarrow 1 < x < 2 and (x + 2) (x - 2) < 0 \Rightarrow - 2 < x < 2 \\ x \in (1, 2) \end{array}$