Summarizo the pertinent information oblained by opplying tho grapting stralegy and sketch the graph of \( y=f(x) \). \[ I(x)=e^{0.5 x}+256 e^{-0.5 x} \] Select the corroct choico below and, if nocessary, fill in the answer box lo complele your choice. A. The \( x \)-intercepl(s) off is \( x= \) \( \square \) (Type an axact answer, using radicals as neoded. Uso a cumma to separato answers as needod.) B. The function thas no \( x \)-intercepts. Solect the correct choice below and, it nocossary, fill in tho answor box to corlpleto your choice. A. The \( y \)-intercopt of I is \( y=257 \) (Type an exact answer, using radicals as neodod.) B. The function \( f \) has no \( y \)-nlercepl. Seloct the correct choice balow and, if necessary, fill in tho answer box to complele your choice. A. The function is increasing on the sutinterval(s) \( \square \) B. The function \( f \) is riever increasing.

The function \( I(x) = e^{0.5x} + 256e^{-0.5x} \) is an interesting one! By analyzing its components, we can find the \( x \)-intercepts by setting \( I(x) = 0 \). Since both parts of the function (\( e^{0.5x} \) and \( 256e^{-0.5x} \)) are always positive for any real \( x \), we can conclude that there are no \( x \)-intercepts. Therefore, the correct answer is: B. The function has no \( x \)-intercepts. To find the \( y \)-intercept, we evaluate the function at \( x = 0 \): \[ I(0) = e^{0.5(0)} + 256e^{-0.5(0)} = 1 + 256 = 257. \] So, the correct answer is: A. The \( y \)-intercept of \( I \) is \( y = 257 \). Lastly, to determine where the function is increasing, we can examine the derivative \( I'(x) \). Since both terms are combinations of exponential functions with positive growth rates, \( I(x) \) is increasing for all \( x \). Thus, the correct answer is: A. The function is increasing on the interval \( (-\infty, \infty) \).