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फलन $f(x)=\frac{\cos ^{-1}\left(\frac{x^2-5 x+6}{x^2-9}\right)}{\log _e\left(x^2-3 x+2\right)}$ का प्रांत है
$(-\infty, 1) \cup(2, \infty)$
$(2, \infty)$
$\left[-\frac{1}{2}, 1\right) \cup(2, \infty)$
$\left[-\frac{1}{2}, 1\right) \cup(2, \infty)-\left\{\frac{3+\sqrt{5}}{2}, \frac{3-\sqrt{5}}{2}\right\}$
Solution
$-1 \leq \frac{x^{2}-5 x+6}{x^{2}-9} \leq 1$
$\frac{x^{2}-5 x+6}{x^{2}-9}-1 \leq 0$
$\frac{1}{x+3} \geq 0$
$x \in(-3, \infty) \ldots \ldots(1)$
$\frac{x^{2}-5 x+6}{x^{2}-9}+1 \geq 0$
$\frac{2 x+1}{x+3} \geq 0$
$x \in(-\infty,-3) \cup\left[-\frac{1}{2}, \infty\right) \ldots \ldots(2)$
after taking intersection
$x \in\left[-\frac{1}{2}, \infty\right)$
$x^{2}-3 x+2>0$
$x \in(-\infty, 1) \cup(2, \infty)$
$x^{2}-3 x+2 \neq 1$
$x \neq \frac{3 \pm \sqrt{5}}{2}$
after taking intersection of each solution
$\left[-\frac{1}{2}, 1\right) \cup(2, \infty)-\left\{\frac{3+\sqrt{5}}{2}, \frac{3-\sqrt{5}}{2}\right\}$
