Find \(g \circ f\) and \(f \circ g\) for Given Relations
📺 Video Explanation
📝 Question
Let
\[ f=\{(3,1),(9,3),(12,4)\} \]
and
\[ g=\{(1,3),(3,3),(4,9),(5,9)\} \]
Show that \(g\circ f\) and \(f\circ g\) are both defined. Also, find \(f\circ g\) and \(g\circ f\).
✅ Solution
🔹 Step 1: Check whether \(g\circ f\) is defined
For \(g\circ f\) to exist, range of \(f\) must be a subset of domain of \(g\).
Domain of \(f\):
\[ \{3,9,12\} \]
Range of \(f\):
\[ \{1,3,4\} \]
Domain of \(g\):
\[ \{1,3,4,5\} \]
Since:
\[ \{1,3,4\}\subseteq\{1,3,4,5\} \]
So, \(g\circ f\) is defined.
🔹 Step 2: Check whether \(f\circ g\) is defined
For \(f\circ g\) to exist, range of \(g\) must be a subset of domain of \(f\).
Range of \(g\):
\[ \{3,9\} \]
Domain of \(f\):
\[ \{3,9,12\} \]
Since:
\[ \{3,9\}\subseteq\{3,9,12\} \]
So, \(f\circ g\) is also defined.
🔹 Step 3: Find \(g\circ f\)
By definition:
\[ (g\circ f)(x)=g(f(x)) \]
- \(f(3)=1 \Rightarrow g(1)=3\)
- \(f(9)=3 \Rightarrow g(3)=3\)
- \(f(12)=4 \Rightarrow g(4)=9\)
Therefore:
\[ g\circ f=\{(3,3),(9,3),(12,9)\} \]
🔹 Step 4: Find \(f\circ g\)
By definition:
\[ (f\circ g)(x)=f(g(x)) \]
- \(g(1)=3 \Rightarrow f(3)=1\)
- \(g(3)=3 \Rightarrow f(3)=1\)
- \(g(4)=9 \Rightarrow f(9)=3\)
- \(g(5)=9 \Rightarrow f(9)=3\)
Therefore:
\[ f\circ g=\{(1,1),(3,1),(4,3),(5,3)\} \]
🎯 Final Answer
\[ \boxed{g\circ f=\{(3,3),(9,3),(12,9)\}} \]
\[ \boxed{f\circ g=\{(1,1),(3,1),(4,3),(5,3)\}} \]
🚀 Exam Shortcut
- For \(g\circ f\): range of \(f\) ⊆ domain of \(g\)
- For \(f\circ g\): range of \(g\) ⊆ domain of \(f\)
- Then evaluate ordered pairs step by step