Find \(g \circ f\) and \(f \circ g\) for \(f(x)=8x^3\) and \(g(x)=\sqrt[3]{x}\)

📺 Video Explanation

📝 Question

Let functions \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be defined as:

\[ f(x)=8x^3,\qquad g(x)=\sqrt[3]{x} \]

Find:

• \((g\circ f)(x)\)
• \((f\circ g)(x)\)

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✅ Solution

🔹 Find \((g\circ f)(x)\)

By definition:

\[ (g\circ f)(x)=g(f(x)) \]

Substitute \(f(x)=8x^3\):

\[ g(8x^3)=\sqrt[3]{8x^3} \]

Now:

\[ \sqrt[3]{8}=2,\qquad \sqrt[3]{x^3}=x \]

So:

\[ (g\circ f)(x)=2x \]

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🔹 Find \((f\circ g)(x)\)

By definition:

\[ (f\circ g)(x)=f(g(x)) \]

Substitute \(g(x)=\sqrt[3]{x}\):

\[ f(\sqrt[3]{x})=8(\sqrt[3]{x})^3 \]

Since:

\[ (\sqrt[3]{x})^3=x \]

Therefore:

\[ (f\circ g)(x)=8x \]

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🎯 Final Answer

\[ \boxed{(g\circ f)(x)=2x} \]

\[ \boxed{(f\circ g)(x)=8x} \]

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🚀 Exam Shortcut

• Cube root and cube cancel each other
• \(\sqrt[3]{8}=2\)
• Always substitute inner function first