Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)=\ln \left(x^{2}+64\right) \) A. The domain is all real \( x \), except \( x=\square \) (Use a comma to separate answers as needed.) B. The domain is all real numbers. Find the \( x \)-intercept(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( x \)-intercept(s) is (are) \( x=\square \) (Round to one decimal place as needed. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts. Find the \( y \)-intercept(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( y \)-intercept(s) is (are) \( y=\square \) (Round to one decimal place as needed. Use a comma to separate answers as needed.) B. There are no \( y \)-intercepts.

**Step 1. Find the domain** For the function \[ f(x)=\ln\left(x^2+64\right), \] the argument of the logarithm must be positive. That is, we require \[ x^2+64 > 0. \] Since \(x^2\) is nonnegative for all real \(x\), and \(64\) is positive, it follows that \[ x^2+64\geq64>0 \] for all real \(x\). Therefore, the domain is: B. The domain is all real numbers. --- **Step 2. Find the \(x\)-intercept(s)** The \(x\)-intercepts are obtained by setting \(f(x)=0\). That gives: \[ \ln\left(x^2+64\right)=0. \] Recall that \(\ln(y)=0\) if and only if \(y=1\). Thus, we set: \[ x^2+64=1. \] Solving for \(x\), we have: \[ x^2 = 1-64 = -63. \] Since \(x^2\) cannot be negative for real numbers, there is no solution. Therefore: B. There are no \(x\)-intercepts. --- **Step 3. Find the \(y\)-intercept(s)** The \(y\)-intercept is found by setting \(x=0\). Substitute into the function: \[ f(0)=\ln\left(0^2+64\right)=\ln(64). \] Numerically, \[ \ln(64) \approx 4.2 \quad (\text{to one decimal place}). \] Thus, the \(y\)-intercept is: A. The \(y\)-intercept(s) is (are) \( y \approx 4.2 \). --- **Summary and Graph Sketch Information** - **Domain:** All real numbers. - **\(x\)-intercepts:** None. - **\(y\)-intercept:** \( (0, \ln(64)) \) or approximately \((0, 4.2)\). For sketching the graph: - The graph of \(\ln\left(x^2+64\right)\) is symmetric about the \(y\)-axis because the expression \(x^2+64\) is even in \(x\). - As \(|x|\) increases, \(x^2+64\) increases, and the logarithm function slowly increases. - The graph has a minimum at \(x=0\) with value \(\ln(64)\). This completes the analysis and sketching strategy for \( f(x)=\ln\left(x^2+64\right) \).