Question

Let $f,g:R\to R$ be functions defined by $f\left(x\right)=\{\begin{array}{cc}\left[x\right],& x<0\\ |1-x|,& x\ge 0\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}{e}^x-x,& x<0\\ (x-1{)}^2-1,x\ge 0& \end{array}$

where $\left[x\right]$ denote the greatest integer less than or equal to $x$. Then, the function $fog$ is discontinuous at exactly :

• A. four points
• B. one point
• C. two points
• D. three points

Answer 1

The correct option is C two points

$f\left(x\right)=\{\begin{array}{cc}\left[x\right],& x<0\\ |1-x|,& x\ge 0\end{array}\text{and g(x)}\{\begin{array}{cc}{e}^x-x,& x<0\\ (x-1{)}^2-1,x\ge 0& \end{array}$

$fog\left(x\right)=\{\begin{array}{cc}\left[g\right(x\left)\right],& g\left(x\right)<0\\ |1-g(x\left)\right|,& g\left(x\right)\ge 0\end{array}$

$\{\begin{array}{cc}|1+x-e^x|,& x<0\\ 1,& x=0\\ \left[\right(x-1{)}^2-1],& 0<x<2\\ |2-(x-1{)}^2|,& x\ge 2\end{array}$

So, $x=0,2$ are the two points where $fog$ is discontinuous.