Let's analyze and evaluate the given piecewise function for the specified values of [tex]\( x \)[/tex]. The function is defined as follows:
[tex]\[
h(x) = \begin{cases}
-5x - 11, & \text{ for } x < -8 \\
3, & \text{ for } -8 \leq x < 3 \\
x + 6, & \text{ for } x \geq 3
\end{cases}
\][/tex]
We need to find the values [tex]\( h(-8) \)[/tex], [tex]\( h(-1) \)[/tex], [tex]\( h(3) \)[/tex], and [tex]\( h(5) \)[/tex].
### 1. [tex]\( h(-8) \)[/tex]
By looking at the definition of the piecewise function, when [tex]\( x = -8 \)[/tex], we need to use the second case:
[tex]\[
-8 \leq x < 3
\][/tex]
Therefore, the value of [tex]\( h(-8) \)[/tex] is:
[tex]\[
h(-8) = 3
\][/tex]
### 2. [tex]\( h(-1) \)[/tex]
To find [tex]\( h(-1) \)[/tex], we check which interval [tex]\(-1\)[/tex] falls into. We see that:
[tex]\[
-8 \leq x < 3
\][/tex]
Thus, the value of [tex]\( h(-1) \)[/tex] is:
[tex]\[
h(-1) = 3
\][/tex]
### 3. [tex]\( h(3) \)[/tex]
Next, for [tex]\( x = 3 \)[/tex], we observe that the third case is applicable because [tex]\( x = 3 \)[/tex] falls into the interval:
[tex]\[
x \geq 3
\][/tex]
Therefore, the value of [tex]\( h(3) \)[/tex] is:
[tex]\[
h(3) = 3 + 6 = 9
\][/tex]
### 4. [tex]\( h(5) \)[/tex]
Lastly, for [tex]\( x = 5 \)[/tex], it falls into the interval:
[tex]\[
x \geq 3
\][/tex]
Thus, the value of [tex]\( h(5) \)[/tex] is:
[tex]\[
h(5) = 5 + 6 = 11
\][/tex]
### Summary
The evaluated values for the function at the given points are:
[tex]\[
h(-8) = 3,\quad h(-1) = 3,\quad h(3) = 9,\quad h(5) = 11
\][/tex]
Thus, the final answer is:
[tex]\[
(3, 3, 9, 11)
\][/tex]
