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 $h_2\left(x\right)$?

• A. Domain is $R$
• B. It is periodic function with period $2\pi$
• C. Range is $[0,1]$
• D. None of these

Answer 1

The correct option is C Range is $[0,1]$

$g\left(x\right)=\sin x,\forall x\in R$

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

$h_2\left(x\right)=|f\left(g\right(x\left)\right)|$ has a domain $R$.


Also, from the graph, it is periodic with period $2\pi$ and has range $[0,2]$.