Question

Find the range of

$f\left(x\right)=|x-1|+|x-2|,-1\le x\le 3$

• A. Range $\in [1,5]$
• B. Range $\in [2,5]$
• C. Range $\in [1,4]$
• D. Range $\in [0,5]$

Answer 1

The correct option is A Range $\in [1,5]$

$f\left(x\right)=|x-1|+|x-2|,-1\le x\le 3$

case 1. $-1\le x\le 1$

$f\left(x\right)=1-x+2-x=3-2x\Rightarrow$ Range $=[1,5]$

case 2. $1<x\le 2$

$f\left(x\right)=x-1+2-x=1\Rightarrow$ Range $=\left\{1\right\}$

case 3. $2<x\le 3$

$f\left(x\right)=x-1+x-2=2x-3\Rightarrow$ Range $=[1,3]$

Hence overall range of the function is, $[1,5]$