A) \[\frac{1}{|x|},x\ne 0\]
B) \[\frac{1}{x}\]for \[|x|>1\]and\[\frac{-1}{x}\]for \[|x|<1\]
C) \[\frac{-1}{x}\]for \[|x|>1\]and \[\frac{1}{x}\]for \[|x|<1\]
D) \[\frac{1}{x}\]for\[x>0\]and\[-\frac{1}{x}\]for \[x<0\]
Correct Answer: B
Solution :
For\[x>1,\]we have \[f(x)=|\log |x||=\log x\to f(x)=\frac{1}{x}\] For\[x<-1,\]we have \[f(x)=|\log |x||=\log (-x)\Rightarrow f(x)=\frac{1}{x}\] For\[0<x<1,\]we have \[f(x)=|\log |x||=-\log x\Rightarrow f(x)=\frac{-1}{x}\] For\[-1<x<0,\]we have \[f(x)=-\log (-x)\Rightarrow f(x)=\frac{1}{x}\] Hence, \[f(x)=\left\{ \begin{matrix} \frac{1}{x}, & |x|>1 \\ -\frac{1}{x}, & |x|<1 \\ \end{matrix} \right.\]
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