Absolutely! Let's go through the step-by-step solution to find the values of the piecewise function [tex]\( h(x) \)[/tex] at specific points: [tex]\( h(-8) \)[/tex], [tex]\( h(-2) \)[/tex], [tex]\( h(2) \)[/tex], and [tex]\( h(8) \)[/tex].
The piecewise function is defined as follows:
[tex]\[ h(x) = \begin{cases}
-5x - 13 & \text{for } x < -8 \\
2 & \text{for } -8 \leq x < 2 \\
x + 3 & \text{for } x \geq 2
\end{cases} \][/tex]
Let's evaluate this step by step:
### 1. Finding [tex]\( h(-8) \)[/tex]:
- According to the piecewise function, the condition for the interval [tex]\( -8 \leq x < 2 \)[/tex] includes [tex]\( x = -8 \)[/tex].
- Thus, when [tex]\( x = -8 \)[/tex]:
[tex]\[ h(-8) = 2 \][/tex]
### 2. Finding [tex]\( h(-2) \)[/tex]:
- Again, the condition for the interval [tex]\( -8 \leq x < 2 \)[/tex] includes [tex]\( x = -2 \)[/tex].
- Therefore, when [tex]\( x = -2 \)[/tex]:
[tex]\[ h(-2) = 2 \][/tex]
### 3. Finding [tex]\( h(2) \)[/tex]:
- For [tex]\( x = 2 \)[/tex], the condition is [tex]\( x \geq 2 \)[/tex].
- Thus, when [tex]\( x = 2 \)[/tex]:
[tex]\[ h(2) = 2 + 3 = 5 \][/tex]
### 4. Finding [tex]\( h(8) \)[/tex]:
- [tex]\( x = 8 \)[/tex] falls into the interval [tex]\( x \geq 2 \)[/tex].
- Therefore, when [tex]\( x = 8 \)[/tex]:
[tex]\[ h(8) = 8 + 3 = 11 \][/tex]
So, summarizing all the results, we have:
[tex]\[ h(-8) = 2 \][/tex]
[tex]\[ h(-2) = 2 \][/tex]
[tex]\[ h(2) = 5 \][/tex]
[tex]\[ h(8) = 11 \][/tex]
Hence, the values for [tex]\( h(-8) \)[/tex], [tex]\( h(-2) \)[/tex], [tex]\( h(2) \)[/tex], and [tex]\( h(8) \)[/tex] are [tex]\( 2 \)[/tex], [tex]\( 2 \)[/tex], [tex]\( 5 \)[/tex], and [tex]\( 11 \)[/tex] respectively.
