Evaluate [tex]\( f(3) \)[/tex] for the piecewise function:

[tex]\[
f(x) =
\begin{cases}
\frac{3x}{2} + 8 & \text{if } x \ \textless \ -6 \\
-3x - 2 & \text{if } -4 \leq x \leq 3 \\
4x + 4 & \text{if } x \ \textgreater \ 3
\end{cases}
\][/tex]

Which value represents [tex]\( f(3) \)[/tex]?

A. -11
B. 8
C. 12.5
D. 16



Answer :

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]