If the domain of the function f(x) = sqrt{ln( | vedclass

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(D) For the function $f(x) = \sqrt{\ln(m\sin x + 4)}$ to be defined for all $x \in R$,the expression inside the square root must be non-negative,i.e.,

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$7$

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(D) For the function $f(x) = \sqrt{\ln(m\sin x + 4)}$ to be defined for all $x \in R$,the expression inside the square root must be non-negative,i.e.,$\ln(m\sin x + 4) \ge 0$.This implies $m\sin x + 4 \ge e^0$,which simplifies to $m\sin x + 4 \ge 1$.Thus,$m\sin x \ge -3$.Since the domain is $R$,this inequality must hold for all $x \in R$. The range of $\sin x$ is $[-1, 1]$.If $m > 0$,the minimum value of $m\sin x$ is $-m$,so $-m \ge -3 \Rightarrow m \le 3$.If $m If $m = 0$,$0 \ge -3$,which is true.Combining these,$m \in [-3, 3]$.The integral values of $m$ are $\{-3, -2, -1, 0, 1, 2, 3\}$.Counting these,there are $7$ possible integral values.

If the domain of the function $f(x) = \sqrt{\ln(m\sin x + 4)}$ is $R$,then the number of possible integral values of $m$ is:

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