The given function $f(x)$ is a piecewise function with three different cases:

1. $f(x) = (x+1)^2 - 1$ for $x < -1$,
2. $f(x) = -x - 2$ for $-1 \leq x \leq 2$,
3. $f(x) = x - 10$ for $x > 2$.

We are tasked with finding $f(-1)$.

Step-by-Step Solution:

1. Identify the case for $x = -1$: Since $-1 \leq x \leq 2$, we use the second case: $f(x) = -x - 2.$

2. Substitute $x = -1$ into $-x - 2$: $f(-1) = -(-1) - 2 = 1 - 2 = -1.$

Final Answer:

$f(-1) = -1.$

Would you like me to explain further or solve additional examples?

Related Questions:

1. How do you evaluate piecewise functions for different values of $x$?
2. What is the value of $f(3)$ for the same piecewise function?
3. Can you graph this piecewise function and describe its behavior?
4. How do you check for continuity at $x = -1$ and $x = 2$?
5. How do piecewise functions model real-world scenarios?

Tip:

Always verify which interval the input value belongs to before substituting into a piecewise function!