Sure, let's analyze the function [tex]\( h(x) \)[/tex] step by step to determine its range.
The function [tex]\( h(x) \)[/tex] is defined piecewise as:
[tex]\[
h(x) = \begin{cases}
x + 2, & \text{if } x < 3 \\
-x + 8, & \text{if } x \geq 3
\end{cases}
\][/tex]
### Analysis of each piece of the function:
1. For [tex]\( x < 3 \)[/tex]:
[tex]\[ h(x) = x + 2 \][/tex]
- As [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex], [tex]\( h(x) = x + 2 \)[/tex] will also approach [tex]\(-\infty\)[/tex].
- When [tex]\( x \)[/tex] is just slightly less than 3, say [tex]\( x = 2.9 \)[/tex], then [tex]\( h(2.9) = 2.9 + 2 = 4.9 \)[/tex] which is close to 5.
- Therefore, for [tex]\( x \)[/tex] just less than 3, [tex]\( h(x) \)[/tex] can get arbitrarily close to 5 from below.
2. For [tex]\( x \geq 3 \)[/tex]:
[tex]\[ h(x) = -x + 8 \][/tex]
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ h(3) = -3 + 8 = 5 \][/tex]
- As [tex]\( x \)[/tex] increases beyond 3, [tex]\( h(x) = -x + 8 \)[/tex] decreases.
- For example, when [tex]\( x = 4 \)[/tex], [tex]\( h(4) = -4 + 8 = 4 \)[/tex], and if [tex]\( x = 5 \)[/tex], [tex]\( h(5) = -5 + 8 = 3 \)[/tex]. This shows a decreasing trend.
Based on these pieces:
### Observing the Range:
- For [tex]\( x < 3 \)[/tex], the function can take values approaching [tex]\(-\infty\)[/tex] as [tex]\( x \)[/tex] decreases without bound.
- For [tex]\( x \geq 3 \)[/tex], the maximum value the function attains is exactly 5 (at [tex]\( x = 3 \)[/tex]) and it decreases as [tex]\( x \)[/tex] increases further.
Therefore, the overall range of [tex]\( h(x) \)[/tex] includes all values less than or equal to 5.
Hence, the correct range of the function [tex]\( h(x) \)[/tex] is:
[tex]\[ \boxed{h(x) \leq 5} \][/tex]
