To determine [tex]\( f(5) \)[/tex] for the given piecewise function
[tex]\[ f(x) = \begin{cases}
x^3 & \text{if } x < -3 \\
2x^2 - 9 & \text{if } -3 \leq x < 4 \\
5x + 4 & \text{if } x \geq 4
\end{cases}, \][/tex]
we need to identify which interval the value [tex]\( x = 5 \)[/tex] falls into and then use the corresponding expression to find [tex]\( f(5) \)[/tex].
1. Determine the interval:
- For the interval [tex]\( x < -3 \)[/tex], [tex]\( 5 \)[/tex] does not fall in this interval.
- For the interval [tex]\( -3 \leq x < 4 \)[/tex], [tex]\( 5 \)[/tex] does not fall in this interval either, because [tex]\( 5 \geq 4 \)[/tex].
- For the interval [tex]\( x \geq 4 \)[/tex], [tex]\( 5 \)[/tex] does fall in this interval.
2. Use the corresponding expression:
Since [tex]\( 5 \geq 4 \)[/tex], we use the expression for the interval [tex]\( x \geq 4 \)[/tex]:
[tex]\[ f(x) = 5x + 4 \][/tex]
3. Substitute [tex]\( x = 5 \)[/tex] into the expression:
[tex]\[ f(5) = 5(5) + 4 \][/tex]
[tex]\[ f(5) = 25 + 4 \][/tex]
[tex]\[ f(5) = 29 \][/tex]
Therefore, the value of [tex]\( f(5) \)[/tex] is [tex]\( \boxed{29} \)[/tex].
