Question

The range of the function $f\left(x\right)=|x-1|+|x-2|,-1\le x\le 3$ is

• A. $[1,3]$
• B. $[1,5]$
• C. $[3,5]$
• D. none of these

Answer 1

The correct option is B $[1,5]$

For $-1\le x<1$

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

$=3-2x$.

For $1\le x<2$

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

$=1$

and for $2\le x<3$

$f\left(x\right)=2x-3$

Hence for $xϵ[-1,3]$

$f(-1)=5$

$f\left(3\right)=3$

So $1$ is the minimum value of $f\left(x\right)$ while $5$ is the maximum value of $f\left(x\right)$.

Hence the range is

$[1,5]$