Question

Let $f\left(x\right)=\left[x\right]$ and $g\left(x\right)=|x|$. Find : $\left(gof\right)(-\frac{5}{3})-\left(fog\right)(-\frac{5}{3})$

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

We have,

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

Domain $\left(f\right)=R$ and, Domain $\left(g\right)=R.$ Therefore, each of $fog,gof$ and $f+2g$ has domain $R.$

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

$=g\left[\right(-\frac{5}{3}\left)\right]-f[|-\frac{5}{3}|]$

$=g(-2)-f\left(\frac{5}{3}\right)=|-2|-\left[\frac{5}{3}\right]=2-1=1$