Find the Required Set
If \( f:[-2,2]\to R \) is defined by
\[ f(x)= \begin{cases} -1, & -2\le x\le0 \\ x-1, & 0\le x\le2 \end{cases} \]
then
\[ \{x\in[-2,2]:x\le0 \text{ and } f(|x|)=x\} \]
is equal to
(a) \(\{-1\}\)
(b) \(\{0\}\)
(c) \(\left\{-\frac12\right\}\)
(d) \(\phi\)
Since \(x\le0\),
\[ |x|=-x\ge0 \]
Therefore,
\[ f(|x|)=|x|-1 \]
Given,
\[ f(|x|)=x \]
\[ |x|-1=x \]
Since \(x\le0\),
\[ -x-1=x \]
\[ -1=2x \]
\[ x=-\frac12 \]
Hence,
\[ \boxed{\left\{-\frac12\right\}} \]
\[ \boxed{\text{Correct Answer: (c)}} \]