Question

The number of points of discontinuity and non-differentiability of the function $f\left(f\right(x\left)\right)$ in its domain are respectively

• A. 2, 2
• B. 1, 1
• C. 1, 2
• D. 2, 1

Answer 1

The correct option is A 2, 2

$f\left(x\right)=\{\begin{array}{cc}x+2,& -2<x<-1\\ -1,& -1\le x<0\\ 0& x=0\\ 1& 0<x<1\\ 1& 1\le x<2\end{array}$

$\Rightarrow f\left(x\right)=\{\begin{array}{cc}x+2,& -2<x<-1\\ -1,& -1\le x<0\\ 0,& x=0\\ 1,& 0<x<2\end{array}$

$f\left(f\right(x\left)\right)=\{\begin{array}{cc}f\left(x\right)+2,& -2<f\left(x\right)<-1\\ -1,& -1\le f\left(x\right)<0\\ 0,& f\left(x\right)=0\\ 1,& 0<f\left(x\right)<2\end{array}$

$f\left(f\right(x\left)\right)=\{\begin{array}{cc}-1,& -1\le x<0\\ 0& x=0\\ 1,& 0<x<2\\ x+2,& -2<x<-1\end{array}$

$\Rightarrow f\left(f\right(x\left)\right)=f\left(x\right)$
Graph of $f\left(x\right)$