$\begin{array}{l} A s s e r t i o n (A) : I f f : R \to R d e f i n e d b y f (x) = \sin^{- 1} \frac{x}{2} + \log_{10} x, t h e n t h e d o m a i n o f t h e f u n c t i o n i s (0, 2] \\ Re a s o n (R) : I f D_{1} a n d D_{2} a r e d o m a i n s o f f_{1} a n d f_{2}, t h e n t h e d o m a i n o f f_{1} a n d f_{2}, t h e n t h e d o m a i n o f f_{1} + f_{2} i s D_{1} \cap D_{2} . \end{array}$

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Both A and R are true and R is the correct explanation of A

A is false but R is true

A is true but R is false

Both A and R are true but R is not correct explanation of A

answer is A.

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Detailed Solution

$\begin{array}{l} A s s e r t i o n : (A) : f (x) = \sin^{- 1} (\frac{x}{2}) + \log_{10}^{x} i s d e f i n e d \\ i f f - 1 \leq \frac{x}{2} \leq 1 a n d x > 0 \\ \Rightarrow - 2 \leq x \leq 2 \\ x \in (0, 2] i s t r u e \\ Re a s o n (R) : t r u e \end{array}$