Questions: Write the domain in interval notation. p(x)=ln(x^2+16) The domain is.

Transcript text: Write the domain in interval notation. \[ p(x)=\ln \left(x^{2}+16\right) \] The domain is $\square$.

Solution

Solution Steps

Step 1: Determine the Argument of the Logarithm

The function $ p(x) = \ln(x^2 + 16) $ requires that the argument $ x^2 + 16 $ be positive. Since $ x^2 $ is always non-negative and $ 16 $ is positive, we have: \[ x^2 + 16 > 0 \quad \text{for all } x \in \mathbb{R} \]

Step 2: Identify the Domain

Since $ x^2 + 16 $ is always positive for all real numbers $ x $, the domain of the function $ p(x) $ is all real numbers. In interval notation, this is expressed as: \[ (-\infty, \infty) \]

Final Answer

The domain of $ p(x) $ is $\boxed{(-\infty, \infty)}$.

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