To find [tex]\( f(0) \)[/tex] and [tex]\( f(9) \)[/tex] using the given piecewise-defined function:
[tex]\[
f(x) =
\begin{cases}
2 - 3x & \text{if } x \leq 3 \\
3x & \text{if } 3 < x < 8 \\
5x + 5 & \text{if } x \geq 8
\end{cases}
\][/tex]
Step-by-Step Solution:
1. Calculate [tex]\( f(0) \)[/tex]:
Determine which piece of the piecewise function [tex]\( x = 0 \)[/tex] falls under:
[tex]\[
x = 0 \quad (\text{since } 0 \leq 3)
\][/tex]
Use the piece of the function [tex]\( f(x) = 2 - 3x \)[/tex]:
[tex]\[
f(0) = 2 - 3(0) = 2
\][/tex]
Therefore, [tex]\( f(0) = 2 \)[/tex].
2. Calculate [tex]\( f(9) \)[/tex]:
Determine which piece of the piecewise function [tex]\( x = 9 \)[/tex] falls under:
[tex]\[
x = 9 \quad (\text{since } 9 \geq 8)
\][/tex]
Use the piece of the function [tex]\( f(x) = 5x + 5 \)[/tex]:
[tex]\[
f(9) = 5(9) + 5 = 45 + 5 = 50
\][/tex]
Therefore, [tex]\( f(9) = 50 \)[/tex].
In conclusion:
[tex]\[
f(0) = 2
\][/tex]
[tex]\[
f(9) = 50
\][/tex]
