Answer :
To evaluate [tex]\( f(9) \)[/tex] from the given piecewise function:
[tex]\[
f(t) =
\begin{cases}
3t^2 + 4 & \text{if } t \leq -4 \\
10 & \text{if } -4 < t \leq 15 \\
1 - 6t & \text{if } t > 15
\end{cases}
\][/tex]
First, determine which piece of the function to use by checking the value of [tex]\( t = 9 \)[/tex]:
1. Check the condition for the first piece [tex]\( (3t^2 + 4, \text{ if } t \leq -4) \)[/tex]:
- Since [tex]\( 9 > -4 \)[/tex], this condition is not satisfied.
2. Check the condition for the second piece [tex]\( (10, \text{ if } -4 < t \leq 15) \)[/tex]:
- Since [tex]\( -4 < 9 \leq 15 \)[/tex], this condition is satisfied. So, we use this piece of the function.
3. Since [tex]\( 9 \)[/tex] falls in the range [tex]\(-4 < t \leq 15\)[/tex], the function simplifies to [tex]\( f(9) = 10 \)[/tex].
Therefore, the value of [tex]\( f(9) \)[/tex] is 10. Thus, the correct answer is [tex]\( \text{b. } 10 \)[/tex].
[tex]\[
f(t) =
\begin{cases}
3t^2 + 4 & \text{if } t \leq -4 \\
10 & \text{if } -4 < t \leq 15 \\
1 - 6t & \text{if } t > 15
\end{cases}
\][/tex]
First, determine which piece of the function to use by checking the value of [tex]\( t = 9 \)[/tex]:
1. Check the condition for the first piece [tex]\( (3t^2 + 4, \text{ if } t \leq -4) \)[/tex]:
- Since [tex]\( 9 > -4 \)[/tex], this condition is not satisfied.
2. Check the condition for the second piece [tex]\( (10, \text{ if } -4 < t \leq 15) \)[/tex]:
- Since [tex]\( -4 < 9 \leq 15 \)[/tex], this condition is satisfied. So, we use this piece of the function.
3. Since [tex]\( 9 \)[/tex] falls in the range [tex]\(-4 < t \leq 15\)[/tex], the function simplifies to [tex]\( f(9) = 10 \)[/tex].
Therefore, the value of [tex]\( f(9) \)[/tex] is 10. Thus, the correct answer is [tex]\( \text{b. } 10 \)[/tex].
