The function $f(x)$ is defined piecewise, meaning that different formulas are used depending on the value of $x$.

Let's analyze the given function.

Definition of the function:

• $f(x) = 2x - 2 - (x - 3)^2 + 6$ for $x \leq -1$
• $f(x) = 11$ for $x \geq 3$

We need to determine which part of the piecewise function to use for $f(0)$.

Since $0$ is not included in the intervals $x \leq -1$ or $x \geq 3$, the function does not provide a definition for $x = 0$. Therefore, $f(0)$ is undefined based on the given information.

If you'd like more clarification or further assistance with a different approach to this problem, feel free to ask.

Relative questions:

1. How do you determine the domain of a piecewise function?
2. What are the steps to evaluate a piecewise function at a specific point?
3. Can the function be redefined to include $x = 0$? If so, how?
4. How would $f(0)$ change if the intervals for $f(x)$ were different?
5. How can you graph a piecewise function based on its definition?

Tip:

When working with piecewise functions, always check which condition the input value satisfies before selecting the corresponding expression.