Question

graph the following function on the axes provided.
$f(x) = \

$$\begin{cases} 2x - 5 & \\text{for} & 1 \\leq x < 5 \\\\ -6 & \\text{for} & x = 5 \\\\ 4 & \\text{for} & x > 5 \\end{cases}$$

$
line closed circle open circle
click and drag to make a line. click the line to delete it.
click on an endpoint of a line to change it.

Explanation:

Step1: Analyze the first piece \( f(x) = 2x - 5 \) for \( 1 \leq x < 5 \)

• Find the endpoints:
• When \( x = 1 \): \( f(1) = 2(1) - 5 = -3 \). Since \( x = 1 \) is included ( \( \leq \) ), we use a closed circle at \( (1, -3) \).
• When \( x = 5 \): \( f(5) = 2(5) - 5 = 5 \). But since \( x = 5 \) is not included in this interval ( \( < 5 \) ), we use an open circle at \( (5, 5) \).
• This is a linear function with slope \( 2 \), so we draw a line segment from \( (1, -3) \) (closed circle) to \( (5, 5) \) (open circle).

Step2: Analyze the second piece \( f(x) = -6 \) for \( x = 5 \)

• This is a single point at \( x = 5 \), \( y = -6 \). So we plot a closed circle at \( (5, -6) \) (since \( x = 5 \) is included here).

Step3: Analyze the third piece \( f(x) = 4 \) for \( x > 5 \)

• For \( x > 5 \), the function is constant \( y = 4 \). We start with an open circle at \( x = 5 \) (since \( x = 5 \) is not included here) at \( (5, 4) \) and draw a horizontal line to the right (for \( x > 5 \)).

To graph:

1. For \( 1 \leq x < 5 \): Draw a line from \( (1, -3) \) (closed circle) to \( (5, 5) \) (open circle) with slope \( 2 \).
2. For \( x = 5 \): Plot a closed circle at \( (5, -6) \).
3. For \( x > 5 \): Draw a horizontal line starting with an open circle at \( (5, 4) \) and extending to the right (for \( x > 5 \)) at \( y = 4 \).

(Note: Since this is a graphing task, the final answer is the graphical representation as described above. If a numerical answer was needed, but in this case, it's a graphing problem with the steps to construct the graph.)

Answer:

The graph consists of:

• A line segment from \((1, -3)\) (closed circle) to \((5, 5)\) (open circle) for \(1 \leq x < 5\).
• A closed circle at \((5, -6)\) for \(x = 5\).
• A horizontal line starting with an open circle at \((5, 4)\) and extending right for \(x > 5\) at \(y = 4\).