Question

If \(f(x) = \frac{4x+3}{6x-4}\), \(x\neq \frac{2}{3}\) and \((\text{fof})(x) = g(x)\) where \(g:R-\{\frac{2}{3}\}\rightarrow R-\{\frac{2}{3}\}\), then \((g\,o\,g\,o\,g\,o\,g\,o\,g)\,(3) =\)

• A. \(3\)
• B. \(\frac{1}{3}\)
• C. \(3^5\)
• D. \(\frac{1}{3^5}\)

Hint:

Compute f o f first; it turns out to be the identity.

Answer 1

The Correct Option is A

Step 1: Show f is self-inverse:
Solve $y = \dfrac{4x+3}{6x-4}$ for $x$: $6xy - 4y = 4x + 3$, so $x(6y - 4) = 4y + 3$ and $x = \dfrac{4y+3}{6y-4}$.

Step 2: Compare:
The inverse has the same formula as $f$, so $f^{-1} = f$ and $f\circ f$ is the identity.

Step 3: Evaluate:
Then $g = $ identity and $g^{5}(3) = 3$, option (A).

Final Answer:
The answer is 3. \[ \boxed{\text{(A) }3} \]