For the functions \( \mathrm{f}(\mathrm{x})=6-\mathrm{x}^{2} \) and \( \mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+4 \mathrm{x}-32 \), find \( \mathrm{f}+\mathrm{g}, \mathrm{f}-\mathrm{g}, \mathrm{fg} \), and \( \frac{\mathrm{f}}{\mathrm{g}} \). Determine the domain for each function. What is the domain of fg ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of fg is \( \{ \) \}. (Use a comma to separate answers as needed.) B. The domain of fg is \( (-\infty, \infty) \). (Type your answer in interval notation.) C. The domain of fg is \( \varnothing \). \( \left(\frac{\mathrm{f}}{\mathrm{g}}\right)(\mathrm{x})=\square \) (Simplify your answer.)

Let’s find each function step by step! 1. **Sum \( f + g \)**: \[ f + g = (6 - x^2) + (x^2 + 4x - 32) = 6 - x^2 + x^2 + 4x - 32 = 4x - 26 \] 2. **Difference \( f - g \)**: \[ f - g = (6 - x^2) - (x^2 + 4x - 32) = 6 - x^2 - x^2 - 4x + 32 = -2x^2 - 4x + 38 \] 3. **Product \( fg \)**: \[ fg = (6 - x^2)(x^2 + 4x - 32) = 6x^2 + 24x - 192 - x^4 - 4x^3 + 32x = -x^4 - 4x^3 + 30x^2 + 24x - 192 \] 4. **Quotient \(\frac{f}{g}\)**: \[ \frac{f}{g} = \frac{6 - x^2}{x^2 + 4x - 32} \] This expression can be simplified but let’s first keep it as is. **Determining the Domain**: - **Domain of \( f \)**: Since \( f(x) = 6 - x^2 \) is a polynomial, its domain is \( (-\infty, \infty) \). - **Domain of \( g \)**: For \( g(x) = x^2 + 4x - 32 \), as it is also a polynomial, the domain is \( (-\infty, \infty) \). **Combining Domains**: - The domain of \( fg \): Since both functions are polynomials, the product's domain is also \( (-\infty, \infty) \). Therefore, you select option **B**. The domain of \( fg \) is \( (-\infty, \infty) \). Finally, the expression for the quotient: \[ \left(\frac{f}{g}\right)(x) = \frac{6 - x^2}{x^2 + 4x - 32} \] This is the final answer! There you have it! 🥳