Animated Solution for Mathematics - Functions: If $f(x)=\frac{4x+3}{6x-4},x\ne \frac{2}{3}$ and $(f\circ f)(x)=g(x)$, where $g:\mathbb{R}-\{\frac{2}{3}\}\to \mathbb{R}-\{\frac{2}{3}\}$, then $(g\circ g\circ g)(4)$ is equal to

Imagine you are standing before a complex, intimidating algebraic expression. You see $f(x)=\frac{4x+3}{6x-4}$ and you are asked to find $(g\circ g\circ g)(4)$, where $g(x)=(f\circ f)(x)$.

It looks like a mountain of calculation, but in the world of JEE Advanced, appearances can be deceiving. Often, the most complex-looking problems are hiding a beautiful, elegant secret.

First, let's define our mission. We are dealing with function composition, where $g(x)=(f\circ f)(x)=f(f(x))$.

We are taking the output of $f(x)$ and feeding it back into $f$ as the new input. To find $g(x)$, we replace every $x$ in the original function with the entire expression:

Let's tackle the numerator first: $4\left(\frac{4x+3}{6x-4}\right)+3$. Using the common denominator $6x-4$, we get:

Now, let's look at the denominator: $6\left(\frac{4x+3}{6x-4}\right)-4$. Again, using the common denominator $6x-4$, we get:

Now, let's bring them together to form $g(x)$:

The denominators $(6x-4)$ are identical, so they cancel out beautifully. We are left with $g(x)=\frac{34x}{34}$, which simplifies to $g(x)=x$.

This is the magic moment! We have discovered that $g(x)$ is the Identity Function. This means $g(g(g(x)))$ is just $x$ composed with itself three times, which remains $x$.

Our goal was to find $(g\circ g\circ g)(4)$. Since $g(x)=x$, then $g(4)=4$, $g(g(4))=4$, and $g(g(g(4)))=4$.

The final answer is 4. All that algebra led us to a clean, simple result, which is the essence of JEE mathematics: finding the underlying structure beneath the noise.