How to Graph the Constant Piecewise Function $f(x)=\begin{cases} 2, x \leq -3 \\ -1, -3

Question

Graph the piecewise function: $f(x)=\begin{cases} 2, \text{ if } x \leq -3 \\ -1, \text{ if } -3 < x < 3 \\ 3, \text{ if } x \geq 3 \end{cases}$

Accepted Answer

A piecewise graph with: 1) A horizontal ray $y=2$ ending at closed point $(-3,2)$ and extending left; 2) A horizontal line segment $y=-1$ with open point $(-3,-1)$ and open point $(3,-1)$; 3) A horizontal ray $y=3$ starting at closed point $(3,3)$ and extending right.

Answer

Step 1: Analyze the first piece $y=2$ for $x \leq -3$: This is a horizontal constant function. Plot the endpoint at $x=-3$: $f(-3)=2$, a closed dot at $(-3,2)$. Draw a horizontal ray from this closed dot extending leftward, staying at $y=2$. Step 2: Analyze the second piece $y=-1$ for $-3 < x < 3$: This is a horizontal constant function. Plot the left endpoint at $x=-3$ as an open dot at $(-3,-1)$ (excluded from the interval) and the right endpoint at $x=3$ as an open dot at $(3,-1)$ (excluded from the interval). Draw a horizontal line segment connecting these two open dots, staying at $y=-1$. Step 3: Analyze the third piece $y=3$ for $x \geq 3$: This is a horizontal constant function. Plot the endpoint at $x=3$: $f(3)=3$, a closed dot at $(3,3)$. Draw a horizontal ray from this closed dot extending rightward, staying at $y=3$. Step 4: Combine all three horizontal segments on the same coordinate grid to form the complete graph.