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To evaluate \( f(-5) \) for the piecewise function provided, we need to determine which part of the function is applicable when \( x = -5 \).

The function is defined as:
- \( f(x) = -4x + 1 \) for \( x < -4 \)
- \( f(x) = -3x \) for \( -4 \leq x \leq 0 \)
- \( f(x) = x^2 - 2 \) for \( 0 \leq x \)

Since \( -5 < -4 \), we use the first case: \( f(x) = -4x + 1 \).

Calculate \( f(-5) \) by substituting \( x = -5 \):
\[
f(-5) = -4(-5) + 1 = 20 + 1 = 21
\]

Thus, \( f(-5) = 21 \).

Do you have any questions about this calculation or need further details on this topic? Here are some related questions you might find helpful:

1. How would you evaluate the function for a value between -4 and 0?
2. Can you explain how the derivative of this piecewise function could be computed?
3. What would be the value of \( f(x) \) if \( x \) were exactly -4?
4. How does the domain of each part of the piecewise function affect its graph?
5. Can you solve for \( x \) given an equation such as \( f(x) = 0 \)?

**Tip:** When working with piecewise functions, always verify which condition your \( x \) value satisfies to choose the correct function expression for evaluation.

Given the following piecewise function, evaluate f(-5).

f(x) =
- -4x + 1 for x < -4
- -3x for -4 ≤ x < 0
- x² - 2 for 0 ≤ x

Learn how to evaluate the piecewise function f(x) when x = -5. The function involves different expressions for different ranges of x, including linear and quadratic parts. This problem is suitable for students in Grades 10-12.

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