1. We are asked to analyze the expression $|x-1| + 1$. 2. The absolute value function $|x-1|$ represents the distance between $x$ and $1$ on the number line. 3. Adding $1$ to $|x-1|$ shifts the output upwards by $1$ unit. 4. We can write the function as $f(x) = |x-1| + 1$. 5. The vertex of the absolute value function $|x-1|$ is at $x=1$, and its value is $0$ there. 6. Therefore, the minimum value of $f(x)$ is at $x=1$, which is $|1-1|+1 = 0 + 1 = 1$. 7. The function is V-shaped with minimum value $1$ at $x=1$. Final answer: The function $f(x) = |x-1| + 1$ has minimum value $1$ at $x=1$.