Question

If $f:R\to R$ and $g:R\to R$ are defined $f\left(x\right)=x-\left[x\right]$ and $g\left(x\right)=\left[x\right]\forall xϵR,f\left(g\right(x\left)\right)$.

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

Answer 1

The correct option is B $0$

We have, $f\left(x\right)=x-\left[x\right]=\left\{x\right\}\dots \left(1\right)$

where $\left\{x\right\}$ is the fractional part of $x$

If $g\left(x\right)=\left[x\right]$, where $\left[x\right]$ is the greatest integer part of $x$, then

The value of $g\left(x\right)$ will always be an integer.

And $\therefore f\left(g\right(x\left)\right)=f\left(\right[x\left]\right)=0$

$\because$ the fractional part of $g\left(x\right)$ i.e $\left[x\right]$ is always $0$.