Question

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

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

Answer 1

Answer: (B)

$0$

Since $f\left(x\right)=x-\left[x\right]$, we have
$f\left[g\right(x\left)\right]=g\left(x\right)-\left[g\right(x\left)\right]$
Also since $g\left(x\right)=\left[x\right]$,we have
$g\left(x\right)-\left[g\right(x\left)\right]=\left[x\right]-\left[\right[x\left]\right]$.
Here, $\left[x\right]$ is the greatest integer not exceeding $x$. Therefore,
$\left[\right[x\left]\right]=\left[x\right]$ and hence,
$\left[x\right]-\left[\right[x\left]\right]=\left[x\right]-\left[x\right]=0$