If f ' ' is the greatest integers function defined on R as f x = x and g ' ' is the modulus function defined…

If ${}^{\text{'}}f^{\text{'}}$ is the greatest integers function defined on $R$ as $f\left(x\right)=\left[x\right]$ and ${}^{\text{'}}g^{\text{'}}$ is the modulus function defined on $R$ as $g\left(x\right)=\left|x\right|$, then the value of $\left(gof\right)\left(\frac{-5}{3}\right)$ is

Given, $g\left(x\right)=\left|x\right|$

$\left(gof\right)\left(\frac{-5}{3}\right)=g\left\{f\left(\frac{-5}{3}\right)\right\}$

$=f\left(\frac{-5}{3}\right)$

$=\left[\frac{-5}{3}\right]$

$=\left[-1.67\right]$

$=-2$     

The Greatest integer less than $-1.67$ is $-2$

But here $\left|x\right|$

Hence, $\left|-2\right|=2$