To determine the range of the step function [tex]\( g(x) \)[/tex], we need to examine how [tex]\( g(x) \)[/tex] is defined for different intervals of [tex]\( x \)[/tex]. Let's analyze the function step by step.
The step function [tex]\( g(x) \)[/tex] is defined as follows:
[tex]\[
g(x)=\left\{\begin{array}{ll}
-3 & \text{if } x \leq 0 \\
2 & \text{if } 0 < x \leq 4 \\
5 & \text{if } 4 < x \leq 10
\end{array}\right.
\][/tex]
From this definition, we can see that:
1. When [tex]\( x \)[/tex] is less than or equal to 0, [tex]\( g(x) = -3 \)[/tex].
2. When [tex]\( x \)[/tex] is greater than 0 and less than or equal to 4, [tex]\( g(x) = 2 \)[/tex].
3. When [tex]\( x \)[/tex] is greater than 4 and less than or equal to 10, [tex]\( g(x) = 5 \)[/tex].
Given this information, [tex]\( g(x) \)[/tex] can take on the values [tex]\(-3\)[/tex], [tex]\(2\)[/tex], and [tex]\(5\)[/tex].
Thus, the range of the function [tex]\( g(x) \)[/tex] is the set of all possible output values, which are [tex]\(-3\)[/tex], [tex]\(2\)[/tex], and [tex]\(5\)[/tex].
Therefore, the correct option is:
[tex]\[ \{-3, 2, 5\} \][/tex]
