A) \[f(\left| x \right|)=\left\{ \begin{matrix} 1-2\le x\le 0 \\ 1-x,0<x\le 2 \\ \end{matrix} \right.\]
B) \[f(\left| x \right|)=x-1\forall x\in R\]
C) \[f(\left| x \right|)=\left\{ \begin{matrix} -x-1,-2\le x\le 0 \\ x-1,0<x\le 2 \\ \end{matrix} \right.\]
D) None of these
Correct Answer: C
Solution :
[c] we have \[f(x)=\left\{ \begin{matrix} -1,-2\le x\le 0 \\ x-1,0\le x\le 2 \\ \end{matrix} \right.\] \[f(\left| x \right|)=\left\{ \begin{matrix} -1,-2\le \left| x \right|\le 0 \\ \left| x \right|-1,0\le \left| x \right|\le 2 \\ \end{matrix} \right.\Rightarrow f(\left| x \right|)=\left| x \right|-1,0\le \left| x \right|\le 2\](\[(as-2\le \left| x \right|\le 0\] is not possible) \[\Rightarrow f(\left| x \right|)=\left\{ \begin{matrix} -x-1, \\ x-1, \\ \end{matrix}\,\,\,\begin{matrix} -2\le x\le 0 \\ 0<x\le 2 \\ \end{matrix}\begin{matrix} {} \\ {} \\ \end{matrix} \right.\]
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