To evaluate \( f(3) \) for the given piecewise function, we need to determine which part of the piecewise function applies when \( x = 3 \). The piecewise function is defined as follows:
[tex]\[
f(x)=\left\{\begin{array}{ll}
\frac{3 x}{2}+8, & x<-6 \\
-3 x-2, & -4 \leq x \leq 3 \\
4 x+4, & x>3
\end{array}\right.
\][/tex]
We need to find the part of the function that is valid for \( x = 3 \):
1. The first piece, \( \frac{3 x}{2} + 8 \), is valid when \( x < -6 \). Since \( 3 \) is not less than \( -6 \), this part does not apply.
2. The second piece, \( -3 x - 2 \), is valid when \( -4 \leq x \leq 3 \). Since \( 3 \) is within this interval (\(-4 \leq 3 \leq 3\)), this part of the function applies.
3. The third piece, \( 4 x + 4 \), is valid when \( x > 3 \). Since \( 3 \) is not greater than \( 3 \), this part does not apply.
Thus, we use the second piece of the function to evaluate \( f(3) \):
[tex]\[
f(x) = -3x - 2 \quad \text{for} \quad -4 \leq x \leq 3
\][/tex]
Substitute \( x = 3 \):
[tex]\[
f(3) = -3(3) - 2
\][/tex]
Now, perform the calculations:
[tex]\[
f(3) = -9 - 2
\][/tex]
[tex]\[
f(3) = -11
\][/tex]
Therefore, the value of the function \( f \) at \( x = 3 \) is:
[tex]\[
f(3) = -11
\][/tex]
