Find the domain of the function $$ f(x)=\operatorname{Ln}\left(4-x^{\wedge} 2\right) $$

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To determine the domain of the function $$ f(x) = \operatorname{Ln}\left(4 - x^2\right) $$, we need to ensure that the argument of the natural logarithm (Ln) is positive because the logarithm is only defined for positive real numbers. ### Step-by-Step Solution: 1. **Set the inside of the logarithm greater than zero:** $$ 4 - x^2 > 0 $$ 2. **Solve the inequality:** $$ -x^2 > -4 \quad \text{(Subtract 4 from both sides)} $$ $$ x^2 < 4 \quad \text{(Multiply both sides by -1, reversing the inequality)} $$ 3. **Find the values of $$ x $$ that satisfy the inequality:** $$ x^2 < 4 \implies -2 < x < 2 $$ ### Conclusion: The domain of $$ f(x) = \operatorname{Ln}\left(4 - x^2\right) $$ consists of all real numbers $$ x $$ for which $$ -2 < x < 2 $$. **Domain:** $$ (-2,\ 2) $$ The domain of the function $$ f(x) = \operatorname{Ln}\left(4 - x^2\right) $$ is all real numbers $$ x $$ such that $$ -2 < x < 2 $$. **Domain:** $$ (-2,\ 2) $$

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