Answer:
See attached for graph.
Step-by-step explanation:
Piecewise functions have multiple pieces of curves/lines where each piece corresponds to its definition over an interval.
Given piecewise function:
[tex]f(x)=\begin{cases}3 & \textsf{if }x\leq -3\\2x+1 & \textsf{if }-3 < x < 4 \\ -2 & \textsf{if } x\geq 4 \end{cases}[/tex]
Therefore, the function has three definitions:
[tex]f(x)=3 \quad \textsf{when x is less than or equal to 3}[/tex]
[tex]f(x)=2x+1 \quad \textsf{when x is more than -3 and less than 4}[/tex]
[tex]f(x)=-2 \quad \textsf{when x is more than or equal to 4}[/tex]
When graphing piecewise functions:
- Use an open circle where the value of x is not included in the interval.
- Use a closed circle where the value of x is included in the interval.
- Use an arrow to show that the function continues indefinitely.
First piece of function
Substitute the endpoint of the interval into the corresponding function:
[tex]\implies f(-3)=3 \implies (-3,3)[/tex]
Place a closed circle at (-3, 3).
As this piece of the function is f(x) = 3 for any value of x that is less than or equal to -3, draw a horizontal straight line to the left from the closed circle. Add an arrow at the end.
Second piece of function
Substitute the endpoints of the interval into the corresponding function:
[tex]\implies f(-3)=2(-3)+1=-5 \implies (-3,-5)[/tex]
[tex]\implies f(4)=2(4)+1=9 \implies (4,9)[/tex]
Place an open circle at (-3, -5) and (4, 9).
Join the points with a straight line.
Third piece of function
Substitute the endpoint of the interval into the corresponding function:
[tex]\implies f(4)=-2 \implies (4,-2)[/tex]
Place a closed circle at (4, -2).
As this piece of the function is f(x) = -2 for any value of x that is more than or equal to 4, draw a horizontal straight line to the right from the closed circle. Add an arrow at the end.
Learn more about piecewise functions here:
https://brainly.com/question/28151925
https://brainly.com/question/11562909
