For \( f(x)=x^{3} \) and \( g(x)=\sqrt[3]{x} \), find \( (f \circ g)(x) \) and \( (g \circ f)(x) \). Then determine whether \( (f \circ g)(x)=(g \circ f)(x) \). What is \( (f \circ g)(x) \) ? \( (f \circ g)(x)=x \) What is \( (g \circ f)(x) \) ? \( (g \circ f)(x)=\square \)

To find \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we need to compose the functions \( f(x) \) and \( g(x) \) in the given order. Given: \( f(x) = x^3 \) \( g(x) = \sqrt[3]{x} \) 1. To find \( (f \circ g)(x) \): - Substitute \( g(x) \) into \( f(x) \): - \( (f \circ g)(x) = f(g(x)) = f(\sqrt[3]{x}) = (\sqrt[3]{x})^3 = x \) 2. To find \( (g \circ f)(x) \): - Substitute \( f(x) \) into \( g(x) \): - \( (g \circ f)(x) = g(f(x)) = g(x^3) = \sqrt[3]{x^3} = x \) Therefore, \( (f \circ g)(x) = x \) and \( (g \circ f)(x) = x \). Since \( (f \circ g)(x) = (g \circ f)(x) = x \), we can conclude that \( (f \circ g)(x) = (g \circ f)(x) \). So, \( (f \circ g)(x) = x \) and \( (g \circ f)(x) = x \).