Question

The range of function $f\left(x\right)=\left|x-1\right|+\left|x-2\right|+\left|x+1\right|+\left|x+2\right|$ where , $x\in \left(-2,2\right)$

• A. 6,8
• B. 2,4
• C. 0,4
• D. 1,2

Answer 1

The correct option is A 6,8

$f\left(x\right)=|x-1|+|x-2|+|x+1|+|x+2|$

$x\in (-2,2)⟹x-2\in (-4,0)$

$⟹x+2\in (0,4)$

$f\left(x\right)=|x-1|+2-x+|x+1|+x+2=4+|x-1|+|x+1|$

$x\in (-2,-1)⟹f\left(x\right)=4+1-x-1-x=4-2x\in (6,8)$

$x\in (-1,1)⟹f\left(x\right)=4+x+1+1-x=6$

$x\in (1,2)⟹f\left(x\right)=4+x-1+x+1=4+2x\in (6,8)$

$\therefore f\left(x\right)\in [6,8]$