Find \((f \circ g)(2)\) and \((g \circ f)(1)\) 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:
- \((f\circ g)(2)\)
- \((g\circ f)(1)\)
✅ Solution
🔹 Find \((f\circ g)(2)\)
By definition:
\[ (f\circ g)(2)=f(g(2)) \]
First find:
\[ g(2)=3(2)^3+1 \]
\[ =3(8)+1=25 \]
Now:
\[ f(25)=25^2+8 \]
\[ =625+8=633 \]
Therefore:
\[ \boxed{(f\circ g)(2)=633} \]
🔹 Find \((g\circ f)(1)\)
By definition:
\[ (g\circ f)(1)=g(f(1)) \]
First find:
\[ f(1)=1^2+8=9 \]
Now:
\[ g(9)=3(9)^3+1 \]
\[ =3(729)+1=2188 \]
Therefore:
\[ \boxed{(g\circ f)(1)=2188} \]
🎯 Final Answer
\[ \boxed{(f\circ g)(2)=633} \]
\[ \boxed{(g\circ f)(1)=2188} \]
🚀 Exam Shortcut
- For \(f\circ g\): first apply \(g\), then \(f\)
- For \(g\circ f\): first apply \(f\), then \(g\)
- Always solve inner function first