Question

Let $f:[-2,2]\to R$ defined by $f\left(x\right)=\{\begin{array}{c}-1,-2\le x<0\\ x-1,0\le x\le 2\end{array}$
then, $\{x|x\in [-2,2]:x\le 0\text{and}f(|x|)=x\}$ is equal to

• A. $\{-1\}$
• B. $\left\{0\right\}$
• C. $\left\{\frac{-1}{2}\right\}$
• D. $\varphi$

Answer 1

The correct option is C $\left\{\frac{-1}{2}\right\}$

We have,
$x\in [-2,2]$
$\therefore |x|\in [0,2]$
$f\left(|x|\right)=x\Rightarrow |x|-1=x$
$\Rightarrow -x-1=x,x<0$
$\Rightarrow 2x=-1$
$\therefore x=-\frac{1}{2}$