Question

Let $f\left(x\right)=sgnx,g\left(x\right)=x(1-x^2)$ then

• A. $fog$ is a continuous function
• B. $gof$ is a continuous function
• C. $fog$ is not continuous at $x=0$
• D. $fog$ is not continuous at $x\in I$

Answer 1

Answer: (B), (C), (D)

$gof$ is a continuous function
$fog$ is not continuous at $x=0$
$fog$ is not continuous at $x\in I$

$f\left(x\right)=sgnx,g\left(x\right)=x(1-x^2)$
We know that signum function is not defined at $x=0$
So, domain of $f\left(x\right)$ is $R-\left\{0\right\}$
$R\left(f\right)=\{-1,0,1\}$
$g\left(x\right)$ is a polynomial function, so its domain and range is R.
Since, $R\left(f\right)\subseteq D\left(g\right)$
$\Rightarrow gof$ is continuous function.
Now, $R\left(g\right)⊈D\left(f\right)$
Hence, $fog$ is not a continuous function.