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Evaluate the step function for the given input values.

[tex]\[
\begin{array}{l}
g(x)=\left\{
\begin{array}{ll}
-4, & -3 \leq x \ \textless \ -1 \\
-1, & -1 \leq x \ \textless \ 2 \\
3, & 2 \leq x \ \textless \ 4 \\
5, & x \geq 4
\end{array}\right. \\
g(2)=\square \\
g(-2)=\square \\
g(5)=\square
\end{array}
\][/tex]


Sagot :

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]