Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)=e^{0.5 x}+256 e^{-0.5 x} \) Select the correct choice below and, if necessery, fill in the answer box to complete your choice. A. The domain of \( f \) is (Type your answer in interval notation. Use a comma to separale answers as needed.) B. The domain of \( f \) is empty.

1. We start with the function \[ f(x)=e^{0.5x}+256\,e^{-0.5x}. \] 2. Both terms, \(e^{0.5x}\) and \(e^{-0.5x}\), are exponential functions. The exponential function \(e^u\) is defined for all \(u \in \mathbb{R}\). Consequently, each term is defined for every real number \(x\). 3. Adding these two functions still results in a function defined for all real numbers. Thus, the domain of \( f \) is all real numbers. 4. In interval notation, the domain is \[ (-\infty,\infty). \] 5. Additional pertinent observations for graphing: - As \( x \to +\infty \), the term \(e^{0.5x}\) dominates, and hence \(f(x) \to \infty\). - As \( x \to -\infty \), the term \(256\,e^{-0.5x}\) dominates, so \(f(x) \to \infty\) as well. - By rewriting the function in the form \(e^{0.5x} + 256\,e^{-0.5x} = u + \frac{256}{u}\) with \(u = e^{0.5x}\), we can observe that the minimum occurs when \(u = \sqrt{256} = 16\). This yields a minimum function value of \[ 16 + \frac{256}{16} = 16 + 16 = 32. \] - The function is U-shaped with a global minimum at \(x = \ln(256)=8\ln2\). 6. Based on the above information, the correct choice is: A. The domain of \( f \) is \((-\infty, \infty)\).