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

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

  1. \varphi(0) = |0-3| + |0-1| = 3 + 1 = 4.
  2. \varphi(1) = |1-3| + |1-1| = 2 + 0 = 2.
  3. \varphi(2) = |2-3| + |2-1| = 1 + 1 = 2.
  4. \varphi(3) = |3-3| + |3-1| = 0 + 2 = 2.
  5. \varphi(-1) = |-1-3| + |-1-1| = |-4| + |-2| = 4+2 = 6.
  6. \varphi(-2) = |-2-3| + |-2-1| = |-5| + |-3| = 5+3 = 8.
  7. 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*}

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