Question

Let $f\left(x\right)=\left[x\right]$ and
$g\left(x\right)=\{\begin{array}{cc}0& \text{if}xisaninteger\\ x^2& otherwise\end{array}$

then

• A. ${\left(gof\right)}^{\prime }\left(1\right)=1$
• B. g of is not continuous
• C. fog is differentiable on $R\sim I$
• D. fog is a differentiable function

Answer 1

The correct option is C fog is differentiable on $R\sim I$

Given $f\left(x\right)=\left[x\right]$
and $g\left(x\right)=\{\begin{array}{cc}0& \text{if}xisaninteger\\ x^2& otherwise\end{array}$
Now, $\left(gof\right)\left(x\right)=g\left[f\right(x\left)\right]$
$=g\left(\right[x\left]\right)$
$=g\left(x\right)$ (Greatest integer function gives an integer )
$=0$
$\Rightarrow \left(gof\right)\left(x\right)=0$
Clearly, option A and B are incorrect.
Now, $\left(fog\right)\left(x\right)=f\left[g\right(x\left)\right]$
$fog\left(x\right)=\{\begin{array}{cc}0& \text{if}xisaninteger\\ n& \text{if}xϵR^{+}and\sqrt{n}<x<\sqrt{n+1}or\\ n& xϵR^{-}and-\sqrt{n+1}<x<-\sqrt{n}\end{array}$
So, fog is differentiable on $R-I$.