The range of fx=sin-1x2+1x2+2 is

Here, x2+1x2+2=1-1x2+2 Now, 2≤x2+2 0or -12≤-1x2+2<0or 12≤1-1x2+2<1or π6≤sin-11-1x2+2<π2

The range of fx=sin-1x2+1x2+2 is

Here, x2+1x2+2=1-1x2+2 Now, 2≤x2+2 0or -12≤-1x2+2<0or 12≤1-1x2+2<1or π6≤sin-11-1x2+2<π2

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The range of $f (x) = \sin^{- 1} (\frac{x^{2} + 1}{x^{2} + 2})$ is

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none of these

$[\frac{π}{6}, \frac{π}{2})$

$[0, \frac{π}{2}]$

$[0, \frac{π}{6}]$

answer is C.

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Detailed Solution

$\begin{array}{l} Here, \frac{x^{2} + 1}{x^{2} + 2} = 1 - \frac{1}{x^{2} + 2} \\ Now, 2 \leq x^{2} + 2 < \infty for all x \in R \end{array}$

$\begin{array}{l} or \frac{1}{2} \geq \frac{1}{x^{2} + 2} > 0 \\ or - \frac{1}{2} \leq \frac{- 1}{x^{2} + 2} < 0 \\ or \frac{1}{2} \leq 1 - \frac{1}{x^{2} + 2} < 1 \\ or \frac{π}{6} \leq \sin^{- 1} (1 - \frac{1}{x^{2} + 2}) < \frac{π}{2} \end{array}$