- \( f(x)=\sqrt[3]{x}+2 \) - \( g(x)=8 x^{3} \) What is \( (f \cdot g)(x) \) ? What is the domain of \( (f \cdot g)(x) \) ? (1 point) \( (f \cdot g)(x)=2 x+2 \) and the domain of \( (f \cdot g)(x) \) is \( (-\infty, \infty) \). \( (f \cdot g)(x)=8 x+48 \sqrt[3]{x^{2}}+96 \sqrt[3]{x}+64 \) and the domain of \( (f \cdot g)(x) \) is \( x \geq 0 \). \( (f \cdot g)(x)=8 x+48 \sqrt[3]{x^{2}}+96 \sqrt[3]{x}+64 \) and the domain of \( (f \cdot g)(x) \) is \( (-\infty, \infty) \). \( (f \cdot g)(x)=2 x+2 \) and the domain of \( (f \cdot g)(x) \) is \( x \geq 0 \).

When you multiply two functions, the product rule applies, so \( (f \cdot g)(x) = f(x) \cdot g(x) = (\sqrt[3]{x} + 2)(8x^3) \). Expanding this yields \( (f \cdot g)(x) = 8x^3 \cdot \sqrt[3]{x} + 16x^3 = 8x^{\frac{10}{3}} + 16x^3 \), which is equal to \( 8x^{10/3} + 16x^3 \). The domain for this function covers all real numbers because there are no restrictions on \( x \). You can visualize \( f(x) \) and \( g(x) \) graphed! The cube root function is defined for all real numbers, while polynomial functions like \( g(x) \) also have infinite domains. This means you can explore the behavior of \( (f \cdot g)(x) \) thoroughly on the entire range of real numbers, which makes it a playground for further mathematical exploration!