Sure, let's carefully evaluate the step function [tex]\( g(x) \)[/tex] for the given inputs.
The step function [tex]\( g(x) \)[/tex] is defined as follows:
[tex]\[
g(x) = \left\{
\begin{array}{ll}
-4, & \text{if } -3 \leq x < -1 \\
-1, & \text{if } -1 \leq x < 2 \\
3, & \text{if } 2 \leq x < 4 \\
5, & \text{if } x \geq 4
\end{array}
\right.
\][/tex]
### Evaluating [tex]\( g(2) \)[/tex]:
We need to determine which interval [tex]\( 2 \)[/tex] falls into:
- Check [tex]\(-3 \leq 2 < -1\)[/tex]: This condition is false.
- Check [tex]\(-1 \leq 2 < 2\)[/tex]: This condition is false.
- Check [tex]\(2 \leq 2 < 4\)[/tex]: This condition is true.
Thus, [tex]\( g(2) \)[/tex] evaluates to 3.
[tex]\[
g(2) = 3
\][/tex]
### Evaluating [tex]\( g(-2) \)[/tex]:
We need to determine which interval [tex]\( -2 \)[/tex] falls into:
- Check [tex]\(-3 \leq -2 < -1\)[/tex]: This condition is true.
Thus, [tex]\( g(-2) \)[/tex] evaluates to -4.
[tex]\[
g(-2) = -4
\][/tex]
### Evaluating [tex]\( g(5) \)[/tex]:
We need to determine which interval [tex]\( 5 \)[/tex] falls into:
- Check [tex]\(-3 \leq 5 < -1\)[/tex]: This condition is false.
- Check [tex]\(-1 \leq 5 < 2\)[/tex]: This condition is false.
- Check [tex]\(2 \leq 5 < 4\)[/tex]: This condition is false.
- Check [tex]\(5 \geq 4\)[/tex]: This condition is true.
Thus, [tex]\( g(5) \)[/tex] evaluates to 5.
[tex]\[
g(5) = 5
\][/tex]
In summary:
[tex]\[
g(2) = 3
\][/tex]
[tex]\[
g(-2) = -4
\][/tex]
[tex]\[
g(5) = 5
\][/tex]
