Summarize the pertinent information obtained by applying the graphing strategy and skelch the graph of \( y=f(x) \). \[ f(x)=e^{0.5 x}+256 e^{-0.5 x} \] Select the correct choice below and, if necossary, fill in the answer box to completo your choice. A. The domain of \( f \) is \( (-\infty, \infty) \). (Type your answer in interval notation. Use a comma lo separate answors as needed.) B. The domain of \( f \) is empty. Select the correct choice below and, if necessary, fill in the answer bux to complete your choice. A. The \( x \)-intercept(s) 0 il is \( x= \) \( \square \) (Type an exact aliswer, using radicals as needed. Use a comma to soparate answers as needed.) B. The function \( f \) has no \( x \)-intercepls Select the correct choice below and, if necessary, Illl in the answer box to complele your choico. A. The \( y \)-intercept of \( f \) is \( y= \) \( \square \) (Type an exact answer, using radicals as needed) B. The function f lias no \( y \)-intercept.

**1. Domain of \( f(x) \)** The function is given by \[ f(x)=e^{0.5x}+256e^{-0.5x}. \] Both \( e^{0.5x} \) and \( 256e^{-0.5x} \) are defined for every real number. Therefore, the domain is \[ (-\infty, \infty). \] **Answer for Domain:** A. The domain of \( f \) is \( (-\infty, \infty) \). --- **2. \( x \)-intercepts** To find an \( x \)-intercept, we set \( f(x)=0 \): \[ e^{0.5x} + 256e^{-0.5x} = 0. \] Notice that for any real \( x \), both \( e^{0.5x} \) and \( 256e^{-0.5x} \) are strictly positive. Their sum is always positive, meaning there is no solution to the equation. **Answer for \( x \)-intercepts:** B. The function \( f \) has no \( x \)-intercepts. --- **3. \( y \)-intercept** The \( y \)-intercept is the value of \( f(0) \): \[ f(0)=e^{0.5\cdot 0}+256e^{-0.5\cdot 0}=1+256=257. \] **Answer for \( y \)-intercept:** A. The \( y \)-intercept of \( f \) is \( y=257 \). --- **Sketching the Graph** - **General Behavior:** - Since both terms in \( f(x) \) are always positive, the graph lies entirely above the \( x \)-axis. - **End Behavior:** - As \( x \to \infty \), \( e^{0.5x} \) grows exponentially while \( e^{-0.5x} \to 0 \). Thus, \[ f(x) \sim e^{0.5x} \quad (\text{increasing to } +\infty). \] - As \( x \to -\infty \), \( e^{0.5x} \to 0 \) and \( e^{-0.5x} \) grows exponentially. Thus, \[ f(x) \sim 256e^{-0.5x} \quad (\text{increasing to } +\infty). \] - **Minimum Point:** - The function has a minimum value since it tends to \( +\infty \) as \( x\to \pm\infty \). Although finding an exact minimum might involve taking the derivative, the pertinent information for the graph includes the presence of a unique minimum somewhere near the \( y \)-intercept. - **Intercepts:** - There is a \( y \)-intercept at \( (0, 257) \). - There are no \( x \)-intercepts. The sketch would show a curve that approaches infinity as \( x \to \pm\infty \) with a single minimum point (the vertex) above the \( x \)-axis, and a \( y \)-intercept at \( (0, 257) \). --- **Final Answers:** - **Domain:** \( (-\infty, \infty) \) - **\( x \)-intercepts:** None - **\( y \)-intercept:** \( y=257 \)