The domain of the function f(x) = sin^{-1}lef | vedclass

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(C) For the function $f(x) = \sin^{-1}(g(x))$ to be defined,the argument $g(x)$ must satisfy $-1 \leq g(x) \leq 1$.Step $1$: Solve $\frac{x^{2}-3x+2}{

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$[-1, \infty)$

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(C) For the function $f(x) = \sin^{-1}(g(x))$ to be defined,the argument $g(x)$ must satisfy $-1 \leq g(x) \leq 1$.Step $1$: Solve $\frac{x^{2}-3x+2}{x^{2}+2x+7} \geq -1$.$x^{2}-3x+2 \geq -(x^{2}+2x+7)$$2x^{2}-x+9 \geq 0$.The discriminant $D = (-1)^{2} - 4(2)(9) = 1 - 72 = -71 Step $2$: Solve $\frac{x^{2}-3x+2}{x^{2}+2x+7} \leq 1$.$x^{2}-3x+2 \leq x^{2}+2x+7$$-3x+2 \leq 2x+7$$-5 \leq 5x \Rightarrow x \geq -1$.Combining both conditions,the domain is $x \in [-1, \infty)$.

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

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