Summarize the pertinent information obtained by applying the graphing strategy and skelch the graph of \( \mathbf{y}=\mathrm{l}(\mathrm{x}) \). \[ f(x)=\left(x^{2}+10\right)\left(100-x^{2}\right) \] Select the correct choico bolow and, if necessary, fill in the answer box lo complete your choice. A. The domain off is \( (-\infty, \infty) \). (Type your answor in interval notetion. Type an exact answar, using radicals as needed. Use a comma to separate answers as noeded.) B. The doman of \( f \) is emply. Select the correct choice below and, if necessary, fill in the answer box in complete your choce. A. The \( x \)-intercepl(s) off is/are \( x= \) \( \square \) (Iype an exact answer, using redicals as ncoded. Use a comma lo separale answers as neoded.) B. The function fhas no \( x \)-ntercepts.

To analyze the function \( f(x) = (x^2 + 10)(100 - x^2) \), we will follow these steps: 1. **Determine the Domain**: - The function \( f(x) \) is a product of two polynomials: \( x^2 + 10 \) and \( 100 - x^2 \). - Since both polynomials are defined for all real numbers, the domain of \( f \) is \( (-\infty, \infty) \). 2. **Find the \( x \)-intercepts**: - The \( x \)-intercepts occur when \( f(x) = 0 \). - Set the function equal to zero: \[ (x^2 + 10)(100 - x^2) = 0 \] - This gives us two equations to solve: 1. \( x^2 + 10 = 0 \) 2. \( 100 - x^2 = 0 \) - For the first equation \( x^2 + 10 = 0 \): - This has no real solutions since \( x^2 = -10 \) is not possible for real \( x \). - For the second equation \( 100 - x^2 = 0 \): - Rearranging gives \( x^2 = 100 \). - Taking the square root gives \( x = 10 \) and \( x = -10 \). Thus, the \( x \)-intercepts are \( x = 10 \) and \( x = -10 \). 3. **Summarize the Findings**: - The domain of \( f \) is \( (-\infty, \infty) \). - The \( x \)-intercepts of \( f \) are \( x = -10 \) and \( x = 10 \). Now, let's fill in the choices based on the findings: ### Domain: A. The domain of \( f \) is \( (-\infty, \infty) \). ### \( x \)-intercepts: A. The \( x \)-intercepts of \( f \) are \( x = -10, 10 \). ### Sketching the Graph: To sketch the graph of \( f(x) \): - The function is a downward-opening parabola due to the \( -x^2 \) term in \( 100 - x^2 \). - The vertex of the parabola can be found by evaluating \( f(0) \): \[ f(0) = (0^2 + 10)(100 - 0^2) = 10 \times 100 = 1000 \] - The graph will have its maximum at \( (0, 1000) \) and will cross the \( x \)-axis at \( (-10, 0) \) and \( (10, 0) \). The graph will look like a downward-opening parabola with the vertex at \( (0, 1000) \) and intercepts at \( (-10, 0) \) and \( (10, 0) \).