Check Whether the Given Function Has an Inverse
📺 Video Explanation
📝 Question
State with reasons whether the following function has inverse:
\[ f:\{1,2,3,4\}\to\{10\} \]
defined by:
\[ f=\{(1,10),(2,10),(3,10),(4,10)\} \]
✅ Solution
🔹 Condition for inverse function
A function has an inverse if and only if it is:
- one-one (injective), and
- onto (surjective)
That is, the function must be bijective.
🔹 Check one-one property
Given:
\[ f(1)=10,\quad f(2)=10,\quad f(3)=10,\quad f(4)=10 \]
Different inputs have the same output:
\[ f(1)=f(2) \quad\text{but}\quad 1\ne2 \]
Therefore:
\[ f \text{ is not one-one} \]
🔹 Check onto property
Codomain is:
\[ \{10\} \]
Since:
\[ f(1)=10 \]
Every element of codomain has a pre-image.
Therefore:
\[ f \text{ is onto} \]
🎯 Final Answer
The function is onto but not one-one.
So, it is not bijective.
Therefore:
\[ \boxed{\text{The function does not have an inverse}} \]
🚀 Exam Shortcut
- Inverse exists only for bijection
- Same output for many inputs ⇒ not one-one
- So inverse does not exist