To evaluate the step function [tex]\( g(x) \)[/tex] for the given input values, we will determine which interval each input value lies in and use the corresponding output value for each interval.
Given the piecewise function:
[tex]\[
g(x)=\left\{
\begin{array}{ll}
-4, & -3 \leq x < -1 \\
-1, & -1 \leq x < 2 \\
3, & 2 \leq x < 4 \\
5, & x \geq 4
\end{array}
\right.
\][/tex]
1. Evaluate [tex]\( g(2) \)[/tex]:
- Since [tex]\( 2 \leq x < 4 \)[/tex], we fall under the interval [tex]\( 2 \leq x < 4 \)[/tex].
- According to the function definition, [tex]\( g(x) = 3 \)[/tex] for [tex]\( 2 \leq x < 4 \)[/tex].
- Therefore, [tex]\( g(2) = 3 \)[/tex].
2. Evaluate [tex]\( g(-2) \)[/tex]:
- Since [tex]\( -3 \leq x < -1 \)[/tex], we fall under the interval [tex]\( -3 \leq x < -1 \)[/tex].
- According to the function definition, [tex]\( g(x) = -4 \)[/tex] for [tex]\( -3 \leq x < -1 \)[/tex].
- Therefore, [tex]\( g(-2) = -4 \)[/tex].
3. Evaluate [tex]\( g(5) \)[/tex]:
- Since [tex]\( x \geq 4 \)[/tex], we fall under the interval [tex]\( x \geq 4 \)[/tex].
- According to the function definition, [tex]\( g(x) = 5 \)[/tex] for [tex]\( x \geq 4 \)[/tex].
- Therefore, [tex]\( g(5) = 5 \)[/tex].
So, the evaluated values are:
[tex]\[
g(2) = 3, \quad g(-2) = -4, \quad g(5) = 5
\][/tex]
In conclusion:
[tex]\[
\begin{array}{l}
g(2) = 3 \\
g(-2) = -4 \\
g(5) = 5
\end{array}
\][/tex]
