Find \(f\circ g\)
📝 Question
Let:
\[ f=\{(1,2),(3,5),(4,1)\} \]
\[ g=\{(2,3),(5,1),(1,3)\} \]
Find \(f\circ g\).
✅ Solution
🔹 Step 1: Definition
\[ (f\circ g)(x)=f(g(x)) \] —
🔹 Step 2: Check domain compatibility
Range of \(g = \{1,3\}\)
Domain of \(f = \{1,3,4\}\)
So composition is defined for inputs where \(g(x)\in\{1,3\}\).
—🔹 Step 3: Compute values
\[ g(2)=3 \Rightarrow f(3)=5 \]
\[ g(5)=1 \Rightarrow f(1)=2 \]
\[ g(1)=3 \Rightarrow f(3)=5 \] —
🔹 Step 4: Write ordered pairs
\[ f\circ g=\{(2,5),(5,2),(1,5)\} \] —
🎯 Final Answer
\[ \boxed{f\circ g=\{(2,5),(5,2),(1,5)\}} \]
🚀 Exam Shortcut
- Apply \(g\) first, then \(f\)
- Match outputs of \(g\) with inputs of \(f\)
- Write final pairs carefully