Show \(g \circ f\) is Defined but \(f \circ g\) is Not Defined
📺 Video Explanation
📝 Question
Let
\[ f=\{(1,-1),(4,-2),(9,-3),(16,4)\} \]
and
\[ g=\{(-1,-2),(-2,-4),(-3,-6),(4,8)\} \]
Show that \(g\circ f\) is defined while \(f\circ g\) is not defined. Also, find \(g\circ f\).
✅ 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 \(g\). :contentReference[oaicite:1]{index=1}
Range of \(f\):
\[ \{-1,-2,-3,4\} \]
Domain of \(g\):
\[ \{-1,-2,-3,4\} \]
Since:
\[ \{-1,-2,-3,4\}\subseteq\{-1,-2,-3,4\} \]
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 \(f\). :contentReference[oaicite:2]{index=2}
Range of \(g\):
\[ \{-2,-4,-6,8\} \]
Domain of \(f\):
\[ \{1,4,9,16\} \]
Since:
\[ \{-2,-4,-6,8\}\nsubseteq\{1,4,9,16\} \]
So, \(f\circ g\) is not defined.
🔹 Step 3: Find \(g\circ f\)
By definition:
\[ (g\circ f)(x)=g(f(x)) \]
- \(f(1)=-1 \Rightarrow g(-1)=-2\)
- \(f(4)=-2 \Rightarrow g(-2)=-4\)
- \(f(9)=-3 \Rightarrow g(-3)=-6\)
- \(f(16)=4 \Rightarrow g(4)=8\)
Therefore:
\[ g\circ f=\{(1,-2),(4,-4),(9,-6),(16,8)\} \]
🎯 Final Answer
\(g\circ f\) is defined, but \(f\circ g\) is not defined.
\[ \boxed{g\circ f=\{(1,-2),(4,-4),(9,-6),(16,8)\}} \]
🚀 Exam Shortcut
- To check \(g\circ f\): compare range of \(f\) with domain of \(g\)
- To check \(f\circ g\): compare range of \(g\) with domain of \(f\)
- Then substitute ordered pairs directly