Question

If $f:[-2,2]\to R$ is defined by $f\left(x\right)=\left\{\begin{array}{c}-1,for-2\le x\le 0then\\ x-1,for0\le x\le 2\end{array}\right\}$

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

Answer 1

The correct option is C

$\{-\frac{1}{2}\}$

Given:
$f\left(x\right)=\left\{\begin{array}{c}-1,for-2\le x\le 0then\\ x-1,for0\le x\le 2\end{array}\right\}$
We know,
$|x|\ge 0$
$\Rightarrow f\left(|x|\right)=|x|-1\dots \dots \left(1\right)$
Also,
If $x\le 0$, then $|x|=-x\dots \dots \left(2\right)\therefore \left\{xϵ\right[-2,2]:x\le 0andf(|x|)=x\}$
$=\{x:|x|-1=x\}$ [Using (1)]
$=\{x:-x-1=x\}$ [Using (2)]
$=\{x:2x=\frac{-1}{2}\}$
$=\{x:x=\frac{-1}{2}\}=\left\{\frac{-1}{2}\right\}$