Shortest Distance from Point to Curve

For $-1 \le x \le 1$, $y = -(x - 1) + (x + 1) = 2$. So the curve contains the horizontal line segment $y = 2$ for $x \in [-1, 1]$. The point $P\left(\frac{1}{2}, 1\right)$ has $x = \frac{1}{2}$, which is in $[-1, 1]$. The perpendicular distance from $\left(\frac{1}{2}, 1\right)$ to the line $y = 2$ is $|2 - 1| = 1$.