Question

f(x) = \

$$\begin{cases} -\\frac{5}{2}x + 6 & \\text{for} & x \\leq 2 \\\\ 3 & \\text{for} & 2 < x \\leq 5 \\\\ 4x - 18 & \\text{for} & x > 5 \\end{cases}$$

line closed circle open circle
click to plot the closed circle.
to delete or draw a line click \line\.

Explanation:

To plot the closed circles for the piecewise function \( f(x) \), we need to find the points where the function has a closed circle (i.e., where the domain includes the endpoint).

Step 1: For \( x \leq 2 \) (function \( -\frac{5}{2}x + 6 \))

The endpoint is \( x = 2 \). Substitute \( x = 2 \) into \( -\frac{5}{2}x + 6 \):
\[
-\frac{5}{2}(2) + 6 = -5 + 6 = 1
\]
So, there is a closed circle at \( (2, 1) \) (since \( x = 2 \) is included in \( x \leq 2 \)).

Step 2: For \( 2 < x \leq 5 \) (function \( 3 \))

The left endpoint \( x = 2 \) is not included (open circle), but the right endpoint \( x = 5 \) is included. Substitute \( x = 5 \) into \( 3 \):
\[
f(5) = 3
\]
So, there is a closed circle at \( (5, 3) \) (since \( x = 5 \) is included in \( 2 < x \leq 5 \)).

Step 3: For \( x > 5 \) (function \( 4x - 18 \))

The endpoint \( x = 5 \) is not included (open circle), so we do not plot a closed circle here.

Final Closed Circles:

• At \( (2, 1) \) (from \( x \leq 2 \))
• At \( (5, 3) \) (from \( 2 < x \leq 5 \))

To plot these:

1. Click “Closed Circle” in the toolbar.
2. Plot \( (2, 1) \) and \( (5, 3) \) on the graph.

(Note: For the lines, draw \( -\frac{5}{2}x + 6 \) for \( x \leq 2 \), a horizontal line \( y = 3 \) for \( 2 < x \leq 5 \), and \( 4x - 18 \) for \( x > 5 \).)

Answer:

To plot the closed circles for the piecewise function \( f(x) \), we need to find the points where the function has a closed circle (i.e., where the domain includes the endpoint).

Step 1: For \( x \leq 2 \) (function \( -\frac{5}{2}x + 6 \))

The endpoint is \( x = 2 \). Substitute \( x = 2 \) into \( -\frac{5}{2}x + 6 \):
\[
-\frac{5}{2}(2) + 6 = -5 + 6 = 1
\]
So, there is a closed circle at \( (2, 1) \) (since \( x = 2 \) is included in \( x \leq 2 \)).

Step 2: For \( 2 < x \leq 5 \) (function \( 3 \))

The left endpoint \( x = 2 \) is not included (open circle), but the right endpoint \( x = 5 \) is included. Substitute \( x = 5 \) into \( 3 \):
\[
f(5) = 3
\]
So, there is a closed circle at \( (5, 3) \) (since \( x = 5 \) is included in \( 2 < x \leq 5 \)).

Step 3: For \( x > 5 \) (function \( 4x - 18 \))

The endpoint \( x = 5 \) is not included (open circle), so we do not plot a closed circle here.

Final Closed Circles:

• At \( (2, 1) \) (from \( x \leq 2 \))
• At \( (5, 3) \) (from \( 2 < x \leq 5 \))

To plot these:

1. Click “Closed Circle” in the toolbar.
2. Plot \( (2, 1) \) and \( (5, 3) \) on the graph.

(Note: For the lines, draw \( -\frac{5}{2}x + 6 \) for \( x \leq 2 \), a horizontal line \( y = 3 \) for \( 2 < x \leq 5 \), and \( 4x - 18 \) for \( x > 5 \).)