First slide

Functions (XII)

Question

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

Moderate

A

$\begin{array}{l} [1, 3] \end{array}$
B

$\begin{array}{l} [1, 5] \end{array}$
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.
$\begin{array}{l} ∴ \max f (x) = the greatest among f (- 1) and f (3) = 5, \\ \min f (x) = f (1) = 1 \end{array}$
So, range = $[1 5]$

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