Find \(g \circ f\) and \(f \circ g\) for \(f(x)=x^2+8\) and \(g(x)=3x^3+1\)
📺 Video Explanation
📝 Question
Let functions \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be defined as:
\[ f(x)=x^2+8,\qquad g(x)=3x^3+1 \]
Find:
- \((g\circ f)(x)\)
- \((f\circ g)(x)\)
✅ Solution
🔹 Find \((g\circ f)(x)\)
By definition:
\[ (g\circ f)(x)=g(f(x)) \]
Substitute \(f(x)=x^2+8\):
\[ g(f(x))=g(x^2+8) \]
Since:
\[ g(x)=3x^3+1 \]
So:
\[ g(x^2+8)=3(x^2+8)^3+1 \]
Therefore:
\[ \boxed{(g\circ f)(x)=3(x^2+8)^3+1} \]
🔹 Find \((f\circ g)(x)\)
By definition:
\[ (f\circ g)(x)=f(g(x)) \]
Substitute \(g(x)=3x^3+1\):
\[ f(3x^3+1) \]
Since:
\[ f(x)=x^2+8 \]
So:
\[ f(3x^3+1)=(3x^3+1)^2+8 \]
Expand:
\[ (3x^3+1)^2=9x^6+6x^3+1 \]
Thus:
\[ (f\circ g)(x)=9x^6+6x^3+9 \]
\[ \boxed{(f\circ g)(x)=9x^6+6x^3+9} \]
🎯 Final Answer
\[ \boxed{(g\circ f)(x)=3(x^2+8)^3+1} \]
\[ \boxed{(f\circ g)(x)=9x^6+6x^3+9} \]
🚀 Exam Shortcut
- \(g\circ f\): substitute whole \(f(x)\) into \(g\)
- \(f\circ g\): substitute whole \(g(x)\) into \(f\)
- Expand only when needed for final simplification