Check Whether the Given Function Has an Inverse
📺 Video Explanation
📝 Question
State with reasons whether the following function has inverse:
\[ g:\{5,6,7,8\}\to\{1,2,3,4\} \]
defined by:
\[ g=\{(5,4),(6,3),(7,4),(8,2)\} \]
✅ Solution
🔹 Condition for inverse function
A function has an inverse if and only if it is bijective.
That means:
- one-one
- onto
🔹 Check one-one property
Given:
\[ g(5)=4,\quad g(6)=3,\quad g(7)=4,\quad g(8)=2 \]
Here:
\[ g(5)=g(7)=4 \]
but:
\[ 5\ne7 \]
Therefore:
\[ g \text{ is not one-one} \]
🔹 Check onto property
Codomain is:
\[ \{1,2,3,4\} \]
Range is:
\[ \{2,3,4\} \]
Element 1 has no pre-image.
Therefore:
\[ g \text{ is not onto} \]
🎯 Final Answer
The function is neither one-one nor onto.
So, it is not bijective.
Therefore:
\[ \boxed{\text{The function does not have an inverse}} \]
🚀 Exam Shortcut
- Check repeated outputs → not one-one
- Check missing codomain values → not onto
- No bijection ⇒ no inverse