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For the piecewise function, find the values [tex]\( h(-8) \)[/tex], [tex]\( h(-1) \)[/tex], [tex]\( h(3) \)[/tex], and [tex]\( h(5) \)[/tex].

[tex]\[
h(x) = \left\{
\begin{array}{ll}
-5x - 11, & \text{for } x \ \textless \ -8 \\
3, & \text{for } -8 \leq x \ \textless \ 3 \\
x + 6, & \text{for } x \geq 3
\end{array}
\right.
\][/tex]


Sagot :

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]
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