Check Whether the Given Function Has an Inverse
📺 Video Explanation
📝 Question
State with reasons whether the following function has inverse:
\[ h:\{2,3,4,5\}\to\{7,9,11,13\} \]
defined by:
\[ h=\{(2,7),(3,9),(4,11),(5,13)\} \]
✅ Solution
🔹 Condition for inverse function
A function has an inverse if and only if it is bijective.
🔹 Check one-one property
Given:
\[ h(2)=7,\quad h(3)=9,\quad h(4)=11,\quad h(5)=13 \]
All outputs are different.
So:
\[ h \text{ is one-one} \]
🔹 Check onto property
Codomain is:
\[ \{7,9,11,13\} \]
Range is:
\[ \{7,9,11,13\} \]
Every element of codomain has a pre-image.
So:
\[ h \text{ is onto} \]
🎯 Final Answer
Since the function is both one-one and onto:
\[ h \text{ is bijective} \]
Therefore:
\[ \boxed{\text{The function has an inverse}} \]
Its inverse is:
\[ \boxed{h^{-1}=\{(7,2),(9,3),(11,4),(13,5)\}} \]
🚀 Exam Shortcut
- Inverse exists only for bijection
- Check all outputs are distinct
- Check every codomain value is used