Summarize the portinent information oblained by applying the graphing sirategy and skelch the graph of \( y=f(x) \) \[ f(x)=\ln (x+8)-1 \] Select the correct choico below and, if necessary, fill in the answer box to complele your choice. A. The domain of \( I \) is \( (-8, \infty) \). (Type your answer in inferval notation. Use a comma to sepanale answers as needed.) B. The domain of \( f \) is emply. Select the cortect choice below and, if necessary, fil in the answet box to complele your cheice A. The \( x \)-niercepi(s) of \( f \) is \( x= \) \( \square \) (Typo an exacl answor Uso a comma lo separalo answers as needed.) B. The function \( f \) has no \( x \)-intercepts.

To graph the function \( f(x) = \ln(x + 8) - 1 \), we first identify the domain. The argument of the logarithm must be greater than zero, leading to \( x + 8 > 0 \) or \( x > -8 \). Thus, the domain of \( f \) is \( (-8, \infty) \). Next, for \( x \)-intercepts, we set \( f(x) = 0 \), resulting in \( \ln(x + 8) - 1 = 0 \). This simplifies to \( \ln(x + 8) = 1 \), leading to \( x + 8 = e \) or \( x = e - 8 \). Since \( e \) is approximately 2.718, \( x \) will be positive, confirming that the function has one \( x \)-intercept at \( x \approx -5.28 \). Another fun fact about logarithmic functions is that they are continuously increasing! So, as \( x \) goes to infinity, \( f(x) \) will also continue to rise without bound, but it will never touch the horizontal line at \( y = -1 \), which represents the horizontal asymptote. Additionally, if you want to take a deeper dive into logarithmic functions, consider exploring how they relate to exponential functions. Understanding the inverse nature of these functions can provide insights into their behavior and applications in fields like finance, biology, and beyond—where growth and decay models are prevalent!