To find [tex]\( f(3) \)[/tex] for the given piecewise function:
[tex]\[
f(x) = \begin{cases}
\frac{3x}{2} + 8 & \text{if } x < -6 \\
-3x - 2 & \text{if } -4 \leq x \leq 3 \\
4x + 4 & \text{if } x > 3
\end{cases}
\][/tex]
we need to determine which piece of the function [tex]\( x = 3 \)[/tex] falls under.
1. The first condition is [tex]\( x < -6 \)[/tex]. Clearly, [tex]\( x = 3 \)[/tex] does not satisfy this condition.
2. The third condition is [tex]\( x > 3 \)[/tex]. Again, [tex]\( x = 3 \)[/tex] does not satisfy this condition.
3. Therefore, [tex]\( x = 3 \)[/tex] falls under the second condition [tex]\( -4 \leq x \leq 3 \)[/tex].
For the second condition, the function is given as:
[tex]\[
f(x) = -3x - 2
\][/tex]
Substitute [tex]\( x = 3 \)[/tex] into the function:
[tex]\[
f(3) = -3(3) - 2
\][/tex]
Calculate the expression:
[tex]\[
-3(3) = -9
\][/tex]
[tex]\[
-9 - 2 = -11
\][/tex]
Therefore, the value of [tex]\( f(3) \)[/tex] is:
[tex]\[
\boxed{-11}
\][/tex]
