Find \(f^{-1}(x)\)
🎥 Video Explanation
📝 Question
Let \( f:\mathbb{R} \to \mathbb{R} \),
\[ f(x)=3x-5 \]
- A. \(\frac{1}{3x-5}\)
- B. \(\frac{x+5}{3}\)
- C. does not exist (not one-one)
- D. does not exist (not onto)
✅ Solution
🔹 Step 1: Check Invertibility
Linear function with non-zero slope ⇒ one-one and onto.
✔️ Inverse exists
—🔹 Step 2: Find Inverse
Let:
\[ y=3x-5 \]
Solve for \(x\):
\[ 3x=y+5 \]
\[ x=\frac{y+5}{3} \] —
🔹 Step 3: Replace \(y\) by \(x\)
\[ f^{-1}(x)=\frac{x+5}{3} \]
—🔹 Final Answer
\[ \boxed{\text{Option B}} \]