Question

Let $\left[x\right]$ denotes the greatest integer $\le x.\mathrm{If}f\left(x\right)=\left[x\right]$ and $g\left(x\right)=\left|x\right|$, then the value of $f\left(g\left(\frac{8}{5}\right)\right)-g\left(f\left(\frac{-8}{5}\right)\right)$ is

• A. $2$
• B. $-2$
• C. $1$
• D. $0$
• E. $-1$

Answer 1

The correct option is E

$-1$

Determine the value of $f\left(g\left(\frac{8}{5}\right)\right)-g\left(f\left(\frac{-8}{5}\right)\right)$.

We have been given:

$f\left(x\right)=\left[x\right]andg\left(x\right)=\left|x\right|$

Now,

$\begin{array}{rcl}f\left(g\left(\frac{8}{5}\right)\right)& =& f\left(\frac{8}{5}\right)\left[\because g\left(x\right)=\left|x\right|\right]\\ & =& \left[\frac{8}{5}\right]\\ & =& 1\left[\because f\left(x\right)=\left[x\right],\mathrm{greatest}\mathrm{integer}<\mathrm{x}\right]\end{array}$

Similarly,

$\begin{array}{rcl}g\left(f\left(\frac{-8}{5}\right)\right)& =& g\left(\left[-\frac{8}{5}\right]\right)\left[\because f\left(x\right)=\left[x\right],\mathrm{greatest}\mathrm{integer}<\mathrm{x}\right]\\ & =& g(-2)\\ & =& |-2|\left[\because g\left(x\right)=\left|x\right|\right]\\ & =& 2\end{array}$

$\begin{array}{rcl}\therefore f\left(g\left(\frac{8}{5}\right)\right)-g\left(f\left(\frac{-8}{5}\right)\right)& =& 1-2\\ & =& -1\end{array}$

Hence the correct option is E.