Question

$f\left(x\right)=sgn\left(x\right),g\left(x\right)=x(x^2-1)$ and $\varphi \left(x\right)=(x^2-1)\sin x,$ then which of the following is/are periodic function

• A. $g\left(f\right(x\left)\right)$
• B. $f\left(g\right(x\left)\right)$
• C. $\varphi \left(f\right(x\left)\right)$
• D. $g\left(\varphi \right(x\left)\right)$

Answer 1

Answer: (A), (C)

The correct options are
A $g\left(f\right(x\left)\right)$
C $\varphi \left(f\right(x\left)\right)$

$f\left(x\right)=1,x>0$

$=-1,x<0$

$=0,x=0$

$g\left(f\right(x\left)\right)=0,\forall x\in R\Rightarrow g\left(f\right(x\left)\right)$ is periodic.

$\varphi \left(f\right(x\left)\right)=0,\forall x\in R\Rightarrow \varphi \left(f\right(x\left)\right)$ is periodic.

But $f\left(g\right(x\left)\right)$and $g\left(\varphi \right(x\left)\right)$ are not periodic.