To find the value of [tex]\( f(-4) + f(3) \)[/tex] for the given piecewise function, we need to evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = -4 \)[/tex] and [tex]\( x = 3 \)[/tex], and then add the results together.
The piecewise function [tex]\( f(x) \)[/tex] is given as:
[tex]\[
f(x) = \begin{cases}
x^2 + 2x - 3 & \text{if } x < -3 \\
4x - 4 & \text{if } -3 \leq x < 6 \\
2x + 2 & \text{if } x \geq 6
\end{cases}
\][/tex]
Step 1: Calculate [tex]\( f(-4) \)[/tex]
Since [tex]\( -4 < -3 \)[/tex], we use the first piece of the function [tex]\( f(x) = x^2 + 2x - 3 \)[/tex].
Substituting [tex]\( x = -4 \)[/tex]:
[tex]\[
f(-4) = (-4)^2 + 2(-4) - 3
\][/tex]
[tex]\[
f(-4) = 16 - 8 - 3
\][/tex]
[tex]\[
f(-4) = 5
\][/tex]
Step 2: Calculate [tex]\( f(3) \)[/tex]
Since [tex]\( -3 \leq 3 < 6 \)[/tex], we use the second piece of the function [tex]\( f(x) = 4x - 4 \)[/tex].
Substituting [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 4(3) - 4
\][/tex]
[tex]\[
f(3) = 12 - 4
\][/tex]
[tex]\[
f(3) = 8
\][/tex]
Step 3: Add [tex]\( f(-4) \)[/tex] and [tex]\( f(3) \)[/tex]
Now, we simply add the results from Step 1 and Step 2:
[tex]\[
f(-4) + f(3) = 5 + 8 = 13
\][/tex]
Therefore, the value of [tex]\( f(-4) + f(3) \)[/tex] is [tex]\( \boxed{13} \)[/tex].
