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