1. **Problem:** Evaluate $f(0)$ given the piecewise function: $$f(x) = \begin{cases} -x + 3 & x > 2 \\ 2x & 0 \leq x \leq 2 \\ 2x - 1 & x < 0 \end{cases}$$ 2. **Step 1:** Identify which piece of the function applies for $x=0$. Since $0 \leq 0 \leq 2$, we use the middle piece: $f(x) = 2x$. 3. **Step 2:** Substitute $x=0$ into $f(x) = 2x$: $$f(0) = 2 \times 0 = 0$$ 4. **Answer:** $$\boxed{f(0) = 0}$$ This means the value of the function $f$ at $x=0$ is 0.