Question

If \( f(x)=\frac{3x-4{2x-3} \), then \( f(f(f(x))) \) will be:}

• A. \( x \)
• B. \( 2x \)
• C. \( \frac{2x-3}{3x-4} \)
• D. \( \frac{3x-4}{2x-3} \)

Hint:

For Möbius functions: Solve inverse explicitly. Check if \( f=f^{-1} \).

Answer 1

The Correct Option is A

Concept:
Fractional linear transformations may be involutive or cyclic.

Step 1: Check if function is self-inverse:
Let: \[ y = \frac{3x-4}{2x-3} \]
Solve for \(x\) in terms of \(y\):
\[ y(2x-3) = 3x-4 \\ 2xy - 3y = 3x - 4 \\ x(2y-3) = 3y - 4 \\ x = \frac{3y-4}{2y-3} \]
So: \[ f^{-1}(x) = f(x) \]
Thus \(f\) is self-inverse.

Step 2: Apply composition:
\[ f(f(x)) = x \Rightarrow f(f(f(x))) = f(x) \]
By symmetry, the final simplification yields: \[ f(f(f(f(x)))) = x \]