Animated Solution for Mathematics - Functions: If $g(f(x))=|\sin x|$ and $f(g(x))=(\sin \sqrt{x}{)}^2$, then

In the world of JEE Advanced, composite functions are essentially pipelines. When we write $g(f(x))$, the input $x$ enters the first machine, $f$, and its output, $f(x)$, becomes the fuel for the second machine, $g$.

We are given the following conditions:

The trap here is to attempt a complex algebraic derivation. Instead, we will utilize the power of strategic verification by testing the proposed functions: $f(x)={\sin }^{2}x$ and $g(x)=\sqrt{x}$.

First, let us calculate $g(f(x))$. We take $f(x)={\sin }^{2}x$ and feed it into $g(x)=\sqrt{x}$.

This yields the expression:

Here is the moment of truth. Many students rush and write $\sin x$, but we must remember that the square root of a square is the absolute value. Therefore:

This matches our first given condition perfectly.

Now, we evaluate the second condition, $f(g(x))$. We take $g(x)=\sqrt{x}$ and feed it into $f(x)={\sin }^{2}x$.

This means we take the sine of the input, $\sqrt{x}$, and then square the result:

This simplifies to:

This matches our second given condition exactly.

We have successfully verified that the functions $f(x)={\sin }^{2}x$ and $g(x)=\sqrt{x}$ satisfy both given equations.

The beauty of this problem lies not in complex derivation, but in understanding the fundamental properties of functions and the efficiency of verification. Keep this mindset, and you will conquer any function problem that comes your way!