Question

If f(x) = sgn (x) and $g\left(x\right)=x(1-x^2)$ then (fog)(x) is discontinuous at

• A. Exactly one point
• B. Exactly two points
• C. Exactly three points
• D. No point.

Answer 1

The correct option is C Exactly three points

$Givenf\left(x\right)=Sgnx=\{\begin{array}{ccc}-1& if& x<0\\ 0& if& x=0\\ 1& if& x>0\end{array}Andg\left(x\right)=x(1-x^2)Nowfog\left(x\right)=-1ifx(1-x^2)<0solving=0ifx(1-x^2)=0,x(1-x^2)<0=1ifx(1-x^2)>0wehavexϵ(-1,0)\cup (1,\infty )\therefore fog\left(x\right)=-1ifxϵ(-1,0)\cup (1,\infty )=0ifxϵ\{-1,0,1\}=1ifxϵ(-\infty ,-1)\cup (0,1)\therefore fog\left(x\right)isdiscontinuousatx=-1,0,1$