First slide

Functions (XII)

Question

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

Moderate

A

$[1, 3]$
B

$[1, 5]$
C

$[3, 5]$
D

none of these

Solution

$Clearly, from the graph, the range is [1, f (- 1)] \equiv [1, 5] .$
$If x < 1, f (x) = - (x - 1) - (x - 2) = - 2 x + 3$
$In this interval, f (x) is decreasing.$
$If 1 \leq x < 2, f (x) = x - 1 - (x - 2) = 1$
$In this interval, f (x) is constant.$
$If 2 \leq x \leq 3, f (x) = x - 1 + x - 2 = 2 x - 3 .$
$In this interval, f (x) is increasing.$
$∴ max f (x) = the greatest among f (- 1) and f (3) = 5$
$min f (x) = f (1) = 1$
$So, range = [1, 5]$

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