Find \(f^{-1}(x)\)
🎥 Video Explanation
📝 Question
Let \( f:\mathbb{R} \to \mathbb{R} \),
\[ f(x)=x-[x] \]
Find \(f^{-1}(x)\).
- (a) \(\frac{1}{x-[x]}\)
- (b) \([x]-x\)
- (c) not defined
- (d) none of these
✅ Solution
🔹 Step 1: Understand Function
\[ f(x)=x-[x] \]
This is the fractional part of \(x\).
\[ f(x)\in[0,1) \]
—🔹 Step 2: Check Injectivity
Many values give same output.
Example:
\[ f(1.2)=0.2,\quad f(2.2)=0.2 \]
❌ Not one-one
—🔹 Step 3: Conclusion
Inverse exists only if function is one-one.
❌ No inverse exists
—🔹 Final Answer
\[ \boxed{\text{Option (c): not defined}} \]