Question

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

$$\begin{cases} -1 & \\text{for } x < -2 \\\\ x - 3 & \\text{for } x > 2 \\end{cases}$$

$
click and drag to make a line. click the line to delete it.
click on an endpoint or a line to change it

Explanation:

Step1: Analyze \( f(x) = -1 \) for \( x < -2 \)

This is a horizontal line. For all \( x \) values less than -2, the \( y \)-value is -1. So we can plot a horizontal line starting from the left (as \( x \) approaches -∞) up to \( x = -2 \) (but not including \( x = -2 \), so we use an open circle at \( (-2, -1) \)).

Step2: Analyze \( f(x) = x - 3 \) for \( x \geq 2 \)

This is a linear function with slope 1 and y-intercept -3. First, find the point at \( x = 2 \): \( f(2) = 2 - 3 = -1 \). So we plot a closed circle at \( (2, -1) \). Then, since the slope is 1, for every 1 unit increase in \( x \), \( y \) increases by 1. So we can draw the line starting from \( (2, -1) \) and going to the right (as \( x \) increases).

Graphing Steps (Summary):

1. For \( x < -2 \): Draw a horizontal line \( y = -1 \) with an open circle at \( (-2, -1) \).
2. For \( x \geq 2 \): Draw the line \( y = x - 3 \) with a closed circle at \( (2, -1) \) and extend it to the right.

(Note: Since this is a text-based explanation, the actual graph would be constructed as described above on the provided axes.)

Answer:

The graph consists of a horizontal line \( y = -1 \) for \( x < -2 \) (open circle at \( (-2, -1) \)) and the line \( y = x - 3 \) for \( x \geq 2 \) (closed circle at \( (2, -1) \)) with slope 1 extending to the right.