Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)=e^{0.5 x}+324 e^{-0.5 x} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of \( f \) is \( (-\infty, \infty) \). (Type your answer in interval notation. Use a comma to separate answers as needed.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The is empty. (Type antercept(s) of \( f \) is \( x=\square \). B. The function \( f \) has no nor \( x \)-intercepts.

The function \( f(x) = e^{0.5x} + 324e^{-0.5x} \) is defined for all real numbers, making its domain \( (-\infty, \infty) \). This means you can plug any value of \( x \) into the function and get a valid output. Now, regarding intercepts, since the function is never equal to zero (as both terms are always positive), it has no \( x \)-intercepts. Another interesting aspect to consider: as \( x \) approaches negative infinity, \( e^{-0.5x} \) dominates and the function goes towards positive infinity. At \( x = 0 \), \( f(0) = 1 + 324 = 325 \). Thus, it has a minimum value at \( x = 0 \) and increases towards positive infinity, emphasizing its balance of growth.