Animated Solution for Mathematics - Functions: Let $g(x)=1+x-[x]$ and $f(x)=\{\begin{array}{ll}-1,& x<0\\ 0,& x=0\\ 1,& x>0\end{array}$. Then for all $x$, $f(g(x))$ is equal to

Unlocking $f(g(x))$

Hello, fellow traveler on the path to JEE excellence! Today, we are going to demystify a problem that often intimidates students: the composition of functions.

When you see $f(g(x))$, it is easy to feel overwhelmed by the piecewise definitions and the greatest integer function. But remember, mathematics is not about memorizing rules; it is about seeing the hidden structure.

Let us break this down together.

Decoding the Inner Function $g(x)$

We are given $g(x)=1+x-[x]$. At first glance, this looks like a standard algebraic expression.

But look closer at the term $x-[x]$. Does it ring a bell? This is the classic definition of the fractional part function, which we denote as $\{x\}$.

So, our function $g(x)$ is simply $1+\{x\}$. This is a massive simplification! Instead of dealing with the greatest integer function directly, we are now working with the fractional part function, which has a beautiful, periodic, sawtooth-like behavior.

The Range Analysis (The "Aha!" Moment)

Now, let us think about the range of $\{x\}$. By definition, the fractional part of any real number $x$ is always greater than or equal to $0$ and strictly less than $1$.

Mathematically, we write this as:

This is the heartbeat of the problem. If we add $1$ to every part of this inequality, we get:

Since $g(x)=1+\{x\}$, we have just discovered that for any real number $x$, the output of $g(x)$ is trapped in the interval $[1,2)$. This is a powerful realization. No matter what value of $x$ you plug into $g(x)$, the result will always be a number between $1$ and $2$.

The Piecewise Bridge

Now, let us look at the outer function $f(x)$. We are given:

In the composite function $f(g(x))$, the output of $g(x)$ becomes the input for $f(x)$. We just established that the output of $g(x)$ is always in the interval $[1,2)$.

Look at this interval carefully. Every single number in $[1,2)$ is strictly positive. This means that the input to $f$ is guaranteed to be greater than $0$ for all $x$.

The Elegance of the Constant

Since the input to $f$ is always strictly positive, we only care about the branch of $f(x)$ where the input is greater than $0$. Looking at the definition, $f(x)=1$ whenever $x>0$.

Therefore, $f(g(x))$ must be equal to $1$ for all real values of $x$. It is a constant function!

We have successfully navigated the complexity and arrived at a simple, elegant result. This is the power of range analysis in composite functions. Keep this technique in your toolkit, and you will find that even the most daunting JEE problems start to yield to your intuition.