Show \(f \circ f(x)=x\) and Find the Inverse of \(f(x)=\frac{4x+3}{6x-4}\)
📺 Video Explanation
📝 Question
Let:
\[ f(x)=\frac{4x+3}{6x-4},\qquad x\ne\frac{2}{3} \]
Show that:
\[ (f\circ f)(x)=x \]
for all:
\[ x\ne\frac{2}{3} \]
Also find:
\[ f^{-1}(x) \]
✅ Solution
🔹 Step 1: Find \(f(f(x))\)
By definition:
\[ (f\circ f)(x)=f\left(\frac{4x+3}{6x-4}\right) \]
Put:
\[ t=\frac{4x+3}{6x-4} \]
Then:
\[ f(t)=\frac{4t+3}{6t-4} \]
Substitute:
\[ f(f(x))= \frac{ 4\left(\frac{4x+3}{6x-4}\right)+3 }{ 6\left(\frac{4x+3}{6x-4}\right)-4 } \]
🔹 Step 2: Simplify numerator
\[ 4\left(\frac{4x+3}{6x-4}\right)+3 \]
Take LCM:
\[ = \frac{16x+12+3(6x-4)}{6x-4} \]
\[ = \frac{16x+12+18x-12}{6x-4} \]
\[ = \frac{34x}{6x-4} \]
🔹 Step 3: Simplify denominator
\[ 6\left(\frac{4x+3}{6x-4}\right)-4 \]
Take LCM:
\[ = \frac{24x+18-4(6x-4)}{6x-4} \]
\[ = \frac{24x+18-24x+16}{6x-4} \]
\[ = \frac{34}{6x-4} \]
🔹 Step 4: Divide
Now:
\[ f(f(x))= \frac{\frac{34x}{6x-4}}{\frac{34}{6x-4}} \]
\[ =x \]
Therefore:
\[ \boxed{(f\circ f)(x)=x} \]
🔹 Step 5: Find inverse
Since:
\[ f(f(x))=x \]
the function is self-inverse.
Hence:
\[ \boxed{f^{-1}(x)=f(x)=\frac{4x+3}{6x-4}} \]
🎯 Final Answer
\[ \boxed{(f\circ f)(x)=x,\quad x\ne\frac{2}{3}} \]
and:
\[ \boxed{f^{-1}(x)=\frac{4x+3}{6x-4}} \]
🚀 Exam Shortcut
- Substitute function into itself
- Simplify numerator and denominator separately
- If \(f\circ f=x\), then function is its own inverse