Question

If $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ and $\mathrm{g}:\mathrm{R}\to \mathrm{R}$ are defined by $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}-\left[\mathrm{x}\right]$ and $\mathrm{g}\left(\mathrm{x}\right)=\left[\mathrm{x}\right]$ for $\mathrm{x}\in \mathrm{R}$, where $\left[\mathrm{x}\right]$ is the greatest integer not exceeding x, then for every $\mathrm{x}\in \mathrm{R}$, $\text{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)$ is equal to

• A. $\mathrm{x}$
• B. $0$
• C. $\mathrm{f}\left(\mathrm{x}\right)$
• D. $\mathrm{g}\left(\mathrm{x}\right)$

Answer 1

The correct option is B

$0$

Explanation for the correct answer:

Since, $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}-\left[\mathrm{x}\right]$, we have

$\mathrm{f}\left(\mathrm{g}\right(\mathrm{x}\left)\right)=\mathrm{g}\left(\mathrm{x}\right)-\left[\mathrm{g}\left(\mathrm{x}\right)\right]$

Also, since $\mathrm{g}\left(\mathrm{x}\right)=\left[\mathrm{x}\right]$, we have

$\mathrm{g}\left(\mathrm{x}\right)-\left[\mathrm{g}\left(\mathrm{x}\right)\right]=\left[\mathrm{x}\right]-\left[\left[\mathrm{x}\right]\right]$

Here, $\left[\mathrm{x}\right]$ is the greatest integer not exceeding x. Therefore,

$\left[\left[\mathrm{x}\right]\right]=\left[\mathrm{x}\right]$

Hence,

$\begin{array}{rcl}\therefore \left[\mathrm{x}\right]-\left[\left[\mathrm{x}\right]\right]& =& \left[\mathrm{x}\right]-\left[\mathrm{x}\right]\\ & =& 0\end{array}$

Hence, the correct answer is an option (B).