https://file.aiquickdraw.com/mathbot/file/1727473593438r3z5txpy.jpg

The image shows a piecewise function defined as follows:

\[
f(x) =
\begin{cases} 
x^2 - 1 & \text{if } -6 \leq x \leq -1 \\
2 - x & \text{if } -1 < x < 2 \\
-x^4 & \text{if } 2 \leq x \leq 6 
\end{cases}
\]

The problem asks to find \( f(0) \).

To solve this, we need to determine which piece of the function applies for \( x = 0 \).

- The first piece applies for \( -6 \leq x \leq -1 \), which does not include \( x = 0 \).
- The second piece applies for \( -1 < x < 2 \), and \( 0 \) lies within this interval.
- The third piece applies for \( 2 \leq x \leq 6 \), which does not include \( x = 0 \).

Thus, \( f(0) \) corresponds to the second piece of the function, which is \( f(x) = 2 - x \).

\[
f(0) = 2 - 0 = 2
\]

Therefore, \( f(0) = 2 \).

Would you like further details or explanations on this solution? Here are 5 related questions you might find interesting:

1. How do you evaluate a piecewise function for different values of \( x \)?
2. What happens if \( x \) lies exactly on the boundaries of two intervals in a piecewise function?
3. Can you graph the given piecewise function to visualize its behavior?
4. How can you determine the continuity of a piecewise function at its boundaries?
5. What is the significance of the piecewise function's different expressions over different intervals?

**Tip:** When working with piecewise functions, always check which interval the input value belongs to before applying the function's rule.

Consider the function f defined piecewise as f(x): x^2 - 1 if -6 ≤ x ≤ -1; 2 - x if -1 < x < 2; -x^4 if 2 ≤ x ≤ 6. Find f(0).

Learn how to evaluate a piecewise function for f(0) where the function is defined by different expressions over specified intervals. This problem covers basic algebra and evaluation of piecewise functions, suitable for Grades 9-12.

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