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 \( (f \circ g)(x) \) ? \( (f \circ g)(x)=\square \)

To find \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we will first define the functions clearly: 1. \( f(x) = x^3 \) 2. \( g(x) = \sqrt[3]{x} \) ### Step 1: Calculate \( (f \circ g)(x) \) The composition \( (f \circ g)(x) \) means we will substitute \( g(x) \) into \( f(x) \): \[ (f \circ g)(x) = f(g(x)) = f\left(\sqrt[3]{x}\right) \] Now substituting \( g(x) \) into \( f(x) \): \[ f\left(\sqrt[3]{x}\right) = \left(\sqrt[3]{x}\right)^3 \] Calculating this gives: \[ \left(\sqrt[3]{x}\right)^3 = x \] Thus, we have: \[ (f \circ g)(x) = x \] ### Step 2: Calculate \( (g \circ f)(x) \) Now, we will calculate \( (g \circ f)(x) \): \[ (g \circ f)(x) = g(f(x)) = g(x^3) \] Substituting \( f(x) \) into \( g(x) \): \[ g(x^3) = \sqrt[3]{x^3} \] Calculating this gives: \[ \sqrt[3]{x^3} = x \] Thus, we have: \[ (g \circ f)(x) = x \] ### Step 3: Compare \( (f \circ g)(x) \) and \( (g \circ f)(x) \) Now we can compare the two results: \[ (f \circ g)(x) = x \] \[ (g \circ f)(x) = x \] Since both compositions yield the same result, we conclude: \[ (f \circ g)(x) = (g \circ f)(x) \] ### Final Answer Thus, the final answer for \( (f \circ g)(x) \) is: \[ (f \circ g)(x) = x \]