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.