To determine [tex]\( f(-2) \)[/tex] for the given piecewise function:
[tex]\[
f(x) = \begin{cases}
x^3 & \text{if } x < -3 \\
2x^2 - 9 & \text{if } -3 \leq x < 4 \\
5x + 4 & \text{if } x \geq 4
\end{cases}
\][/tex]
We will follow these steps:
1. Identify which piece of the function applies to the value [tex]\( x = -2 \)[/tex].
2. Evaluate the function using the specific piece that applies.
### Step-by-Step Solution:
1. Determine the correct interval for [tex]\( x = -2 \)[/tex]:
- [tex]\( -2 \)[/tex] is not less than [tex]\(-3\)[/tex], so the first piece [tex]\( x^3 \)[/tex] does not apply.
- [tex]\( -2 \)[/tex] is within the interval [tex]\([-3, 4)\)[/tex]. Therefore, the second piece [tex]\( 2x^2 - 9 \)[/tex] applies to [tex]\(-2\)[/tex].
2. Evaluate the correct piece [tex]\( 2x^2 - 9 \)[/tex] at [tex]\( x = -2 \)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] into the expression [tex]\( 2x^2 - 9 \)[/tex]:
[tex]\[
f(-2) = 2(-2)^2 - 9
\][/tex]
3. Calculate the value:
- Compute [tex]\((-2)^2\)[/tex]:
[tex]\[
(-2)^2 = 4
\][/tex]
- Multiply [tex]\(4\)[/tex] by [tex]\(2\)[/tex]:
[tex]\[
2 \times 4 = 8
\][/tex]
- Subtract [tex]\(9\)[/tex] from [tex]\(8\)[/tex]:
[tex]\[
8 - 9 = -1
\][/tex]
Thus, the value of [tex]\( f(-2) \)[/tex] is [tex]\(-1\)[/tex].
Therefore, the answer is [tex]\( \boxed{-1} \)[/tex].
