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The domain of the function $$f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)$$ is :

JEE · Math · previous-year question

  1. A.$$[1, \infty)$$
  2. B.$$[-1,2]$$
  3. C.$$[-1, \infty)$$correct
  4. D.$$(-\infty, 2]$$

Answer

C. $$[-1, \infty)$$

Explanation

$f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)$ $$ -1 \leq \frac{x^{2}-3 x+2}{x^{2}+2 x+7} \leq 1 $$ $$ \begin{aligned} & \frac{x^{2}-3 x+2}{x^{2}+2 x+7} \leq 1 \\\\ & x^{2}-3 x+2 \leq x^{2}+2 x+7 \\\\ & 5 x \geq-5 \\\\ & x \geq-1 \end{aligned} $$ And $$ \frac{x^{2}-3 x+2}{x^{2}+2 x+7} \geq-1 $$ $$ x^{2}-3 x+2 \geq-x^{2}-2 x-7 $$ $2 x^{2}-x+9 \geq 0$ $$ x \in R $$ (i) $\cap$ (ii) Domain $\in[-1, \infty)$

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