First slide

Introduction to real valued functions

Question

The range of $| x - 2 | + | x - 5 |$ is

Moderate

A

$[2, \infty)$
B

$[3, \infty)$
C

$[4, \infty)$
D

$[5, \infty)$

Solution

$f (x) = | x - 2 | + | x - 5 |$ and domain f = R
For $x < 2, f (x) = 2 - x + 5 - x = 7 - 2 x > 3;$
For $2 < x < 5 f (x) = x - 2 + 5 - x = 3$
For $x > 5, f (x) = x - 2 + x - 5 = 2 x - 7 > 3$
Range $f = [3, \infty)$
$(0 R) R a n g e o f |x - a| + |x - b| i s [|b - a|, \infty)$ $= [|5 - 2|, \infty) = [3, \infty)$

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