Let \( f(x)=2 x^{3} \) and \( g(x)=\sqrt[3]{x} \). Find \( (f g)(x) \) and \( \left(\frac{f}{g}\right)(x) \). Then evaluate \( f g \) and \( \begin{array}{l}(f g)(x)=\square ;(f g)(-27)=118096 \\ \left(\frac{f}{g}\right)(x)=\square ;\left(\frac{f}{g}\right)(-27)=13122 \\ \text { State the domain of }(f g)(x) \text { and }\left(\frac{f}{g}\right)(x) \\ \text { The domain of }(f g)(x) \text { is }\end{array} \) 8.

Let's solve the problem step by step. Given: - \( f(x) = 2x^3 \) - \( g(x) = \sqrt[3]{x} = x^{1/3} \) ### 1. Finding \( (f g)(x) \) Assuming \( (f g)(x) \) represents the **product** of \( f(x) \) and \( g(x) \): \[ (f g)(x) = f(x) \cdot g(x) = 2x^3 \cdot x^{1/3} = 2x^{3 + \frac{1}{3}} = 2x^{\frac{10}{3}} \] **Evaluating at \( x = -27 \):** \[ (f g)(-27) = 2(-27)^{\frac{10}{3}} \] First, compute \( (-27)^{1/3} = -3 \), then raise it to the 10th power: \[ (-3)^{10} = 59049 \] So, \[ (f g)(-27) = 2 \times 59049 = 118098 \] ### 2. Finding \( \left(\frac{f}{g}\right)(x) \) \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{2x^3}{x^{1/3}} = 2x^{3 - \frac{1}{3}} = 2x^{\frac{8}{3}} \] **Evaluating at \( x = -27 \):** \[ \left(\frac{f}{g}\right)(-27) = 2(-27)^{\frac{8}{3}} \] First, compute \( (-27)^{1/3} = -3 \), then raise it to the 8th power: \[ (-3)^8 = 6561 \] So, \[ \left(\frac{f}{g}\right)(-27) = 2 \times 6561 = 13122 \] ### 3. Determining the Domains - **Domain of \( (f g)(x) \):** Since both \( f(x) = 2x^3 \) and \( g(x) = \sqrt[3]{x} \) are defined for all real numbers \( x \), their product is also defined for all real numbers. \[ \text{Domain of } (f g)(x) \text{ is } \mathbb{R} \] - **Domain of \( \left(\frac{f}{g}\right)(x) \):** For the quotient \( \frac{f(x)}{g(x)} \) to be defined, \( g(x) \) must not be zero. Since \( g(x) = \sqrt[3]{x} \) is zero only when \( x = 0 \), we exclude this point from the domain. \[ \text{Domain of } \left(\frac{f}{g}\right)(x) \text{ is } \mathbb{R} \setminus \{0\} \] ### Summary of Results \[ \begin{align*} (f g)(x) &= 2x^{10/3} \\ (f g)(-27) &= 118098 \\ \left(\frac{f}{g}\right)(x) &= 2x^{8/3} \\ \left(\frac{f}{g}\right)(-27) &= 13122 \\ \text{Domain of } (f g)(x) &= \mathbb{R} \\ \text{Domain of } \left(\frac{f}{g}\right)(x) &= \mathbb{R} \setminus \{0\} \end{align*} \]