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 necessary, fill in the answer box to complele your choice. A. The domain of \( f \) is \( (-\infty, \infty) \). (Type your answer in interval notation. Use a comma to separate answers as needed.) Belect the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( x \)-intercept(s) of \( f \) is \( x=\square \). (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. The function \( f \) has no \( x \)-intercepts.
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The domain of the function \( f(x) = e^{0.5x} + 256e^{-0.5x} \) is indeed all real numbers, which we can express in interval notation as \( (-\infty, \infty) \). This means that you can plug in any real number for \( x \) and get a corresponding \( f(x) \) value. Now, to find the \( x \)-intercept(s), we set \( f(x) = 0 \). Solving the equation \( e^{0.5x} + 256e^{-0.5x} = 0 \) leads us to realize that both \( e^{0.5x} \) and \( 256e^{-0.5x} \) are always positive, so there are no values of \( x \) that satisfy this equation. Therefore, the function \( f \) has no \( x \)-intercepts, which is reflected in choice B.
