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

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(C) The domain of $\sin^{-1}(u)$ is $u \in [-1, 1]$.Therefore,we must have $-1 \le 1 + 3x + 2x^2 \le 1$.Case $I$: $2x^2 + 3x + 1 \ge -1$$2x^2 + 3x + 2

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$\left[ - \frac{3}{2}, 0 \right]$

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(C) The domain of $\sin^{-1}(u)$ is $u \in [-1, 1]$.Therefore,we must have $-1 \le 1 + 3x + 2x^2 \le 1$.Case $I$: $2x^2 + 3x + 1 \ge -1$$2x^2 + 3x + 2 \ge 0$.The discriminant $D = 3^2 - 4(2)(2) = 9 - 16 = -7 Case $II$: $2x^2 + 3x + 1 \le 1$$2x^2 + 3x \le 0$$x(2x + 3) \le 0$.The roots are $x = 0$ and $x = -\frac{3}{2}$.Testing the intervals,the inequality holds for $x \in \left[ -\frac{3}{2}, 0 \right]$.Combining both cases,the domain is $\left[ -\frac{3}{2}, 0 \right]$.

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

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