JEE Main 2024 — Functions Question with Solution

To find $$(g\circ g\circ g)(4),$$ we first need to understand the composition of $$f$$ with itself, i.e., $$(f\circ f)(x)=f(f(x))=g(x).$$ We can then repeatedly apply $$g$$ to get the given expression.

First, let's calculate $$(f\circ f)(x)=g(x):$$

$$g(x)=(f\circ f)(x)=f(f(x))$$

$$=f\left(\frac{4x+3}{6x-4}\right)$$

To evaluate this expression, we substitute $$\frac{4x+3}{6x-4}$$ for $$x$$ in the function $$f(x):$$

$$g(x)=f\left(\frac{4x+3}{6x-4}\right)=\frac{4\left(\frac{4x+3}{6x-4}\right)+3}{6\left(\frac{4x+3}{6x-4}\right)-4}$$

Now, we simplify the expression:

$$g(x)=\frac{4(4x+3)+3(6x-4)}{6(4x+3)-4(6x-4)}$$

$$=\frac{16x+12+18x-12}{24x+18-24x+16}$$

$$=\frac{34x}{34}$$

$$=x$$

So, $$g(x)=x$$ for all $$x$$ in the domain of $$g$$, which is $$\mathbb{R}-\left\{\frac{2}{3}\right\}$$. It's important to note that the domain restriction is preserved through the composition because $$f(x)$$ has a vertical asymptote at $$x=\frac{2}{3}$$ which doesn't intersect the graph.

So, $$g(x)$$ is the identity function on its domain, which means that applying $$g$$ any number of times will result in the same input for $$x$$ in the given domain. Hence, we have:

$$(g\circ g\circ g\circ g)(4)=g(g(g(g(4))))=g(g(g(4)))=g(g(4))=g(4)=4$$

This corresponds to option D, which is $$4$$.