Find \(f^{-1}(1)\)
🎥 Video Explanation
📝 Question
Let \( f:\mathbb{R} \to \mathbb{R} \),
\[ f(x)=\tan x \]
Find \(f^{-1}(1)\).
- (a) \(\frac{\pi}{4}\)
- (b) \(\{n\pi+\frac{\pi}{4}: n\in\mathbb{Z}\}\)
- (c) does not exist
- (d) none of these
✅ Solution
🔹 Step 1: Check Invertibility
\(\tan x\) is periodic:
\[ \tan x = \tan(x+n\pi) \]
So it is not one-one on \(\mathbb{R}\).
❌ No inverse function exists on \(\mathbb{R}\)
—🔹 Step 2: Meaning of \(f^{-1}(1)\)
Since inverse does not exist as a function,
\(f^{-1}(1)\) is not defined.
—🔹 Final Answer
\[ \boxed{\text{Option (c): does not exist}} \]