Question

$f\left(x\right)=\{\begin{array}{cc}x-1,& -1\le x<0\\ x^2,& 0\le x\le 1\end{array}$ and $g\left(x\right)=\sin x$. Consider the function $h_1\left(x\right)=f\left(|g\right(x\left)|\right)$ and $h_2\left(x\right)=|f\left(g\right(x\left)\right)|$
Which of the following is not true about $h_1\left(x\right)$?

• A. It is periodic function with period $\pi$
• B. Range is $[0,1]$
• C. Domain is $R$
• D. None of these

Answer 1

The correct option is D None of these

$|g\left(x\right)|=|\sin x|,\forall x\in R$
$f\left(|g\right(x\left)|\right)=\{\begin{array}{cc}|\sin x|-1,& -1\le |\sin x|<0\\ (|\sin x|{)}^2,& 0\le |\sin x|\le 1\end{array}\Rightarrow f\left(|g\right(x\left)|\right)={\sin }^{2}x$

Clearly, $h_1\left(x\right)=f\left(|g\right(x\left)|\right)={\sin }^{2}x$ has a period $\pi$, range $[0,1]$, and domain $R$