Question

Consider the function $f\left(x\right)=sgnx$ and $g\left(x\right)=x(1-x^2)$. Which of the following does NOT hold good?

• A. $\left(fog\right)\left(x\right)$ is neither odd nor even
• B. $\left(gof\right)\left(x\right)$ is odd
• C. $\left(fog\right)\left(x\right)$ is neither continuous nor differentiable for some $x$ on $(-\infty ,\infty )$
• D. $\left(fog\right)\left(x\right)$ is continuous and differentiable for every $x$ on $(-\infty ,\infty )$

Answer 1

Answer: (A), (C)

The correct options are
A $\left(fog\right)\left(x\right)$ is neither odd nor even
C $\left(fog\right)\left(x\right)$ is neither continuous nor differentiable for some $x$ on $(-\infty ,\infty )$

$\text{sgn}\left(x\right)=\{\begin{array}{cc}-1& x<0\\ 0& x=0\\ 1& x>0\end{array}$
$g\left(x\right)=x(1-x^2)$
$=x(1-x)(1+x)$
$g\left(x\right)=0$ implies, $x=\{-1,0,1\}$.
$\text{sgn}\left(g\right(x\left)\right)=\{\begin{array}{cc}-1& x\in (-\infty ,-1)\cup (1,\infty )\\ 0& x=\{-1,0,1\}\\ 1& x\in (-1,1)\end{array}$