Find the domain of the logarithmic function.

The domain of is . (Type your answer in interval notation.)

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Mar 23,2025

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Find the domain of the logarithmic function. $$ f(x)=\ln \left(x^{2}-x-6\right) $$ The domain of $$ f $$ is $$ \square $$. (Type your answer in interval notation.)

UpStudy AI · Accepted Answer

1. To find the domain, set the argument of the logarithm greater than zero: $$ x^{2} - x - 6 > 0 $$ 2. Factor the quadratic: $$ x^{2} - x - 6 = (x - 3)(x + 2) $$ 3. Set up the inequality: $$ (x - 3)(x + 2) > 0 $$ 4. Identify the critical points where the expression equals zero: $$ x - 3 = 0 \quad \Rightarrow \quad x = 3 $$ $$ x + 2 = 0 \quad \Rightarrow \quad x = -2 $$ 5. Test intervals determined by the critical points: - For $$ x < -2 $$, choose $$ x = -3 $$: $$ (-3 - 3)(-3 + 2) = (-6)(-1) = 6 > 0 $$ - For $$ -2 < x < 3 $$, choose $$ x = 0 $$: $$ (0 - 3)(0 + 2) = (-3)(2) = -6 < 0 $$ - For $$ x > 3 $$, choose $$ x = 4 $$: $$ (4 - 3)(4 + 2) = (1)(6) = 6 > 0 $$ 6. The inequality holds true for: $$ x < -2 \quad \text{or} \quad x > 3 $$ 7. Thus, the domain of the function, in interval notation, is: $$ (-\infty, -2) \cup (3, \infty) $$ The domain of $$ f(x) = \ln(x^{2} - x - 6) $$ is $$ (-\infty, -2) \cup (3, \infty) $$.

Find the domain of the logarithmic function.

The domain of is . (Type your answer in interval notation.)

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Find the domain of the logarithmic function. The domain of is . (Type your answer in interval notation.)