Compute some values of phi(x) = |x-3| + |x-1|

by RoRi

July 11, 2015

Define $\varphi(x) = |x-3| + |x-1|$ for all $x \in \mathbb{R}$ . Compute:

$\varphi(0) = |0-3| + |0-1| = 3 + 1 = 4$ .
$\varphi(1) = |1-3| + |1-1| = 2 + 0 = 2$ .
$\varphi(2) = |2-3| + |2-1| = 1 + 1 = 2$ .
$\varphi(3) = |3-3| + |3-1| = 0 + 2 = 2$ .
$\varphi(-1) = |-1-3| + |-1-1| = |-4| + |-2| = 4+2 = 6$ .
$\varphi(-2) = |-2-3| + |-2-1| = |-5| + |-3| = 5+3 = 8$ .
Find all $t \in \mathbb{R}$ such that $\varphi(t+2) = \varphi(t)$ .
$\begin{align*} &&\varphi(t+2) &= \varphi(t) \\ \implies && |t+2 - 3| + |t+2-1| &= |t-3|+|t-1| \\ \implies && |t-1| + |t+1| &= |t-3| + |t-1| \\ \implies && |t+1| &= |t-3| \implies && t = 1 \end{align*}$