Question

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

$$\begin{cases}-2&\\text{for}&-4\\leq x\\leq -1\\\\x - 2&\\text{for}&-1 < x\\leq 3\\end{cases}$$

$
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: Graph the first piece

The first piece of the function is \( f(x) = -2 \) for \( -4 \leq x \leq -1 \). This is a horizontal line. We plot the points \( (-4, -2) \) and \( (-1, -2) \) (since the endpoints are included, we use closed circles) and draw a horizontal line segment between them.

Step2: Graph the second piece

The second piece is \( f(x) = x - 2 \) for \( -1 < x \leq 3 \). First, find the value at \( x = -1 \) (open circle, since \( x = -1 \) is not included here: \( f(-1) = -1 - 2 = -3 \), but we use an open circle at \( (-1, -3) \)). Then find the value at \( x = 3 \) (closed circle, since \( x = 3 \) is included: \( f(3) = 3 - 2 = 1 \), so we plot \( (3, 1) \)). We can also find another point, for example, at \( x = 0 \), \( f(0) = 0 - 2 = -2 \), so \( (0, -2) \); at \( x = 1 \), \( f(1) = 1 - 2 = -1 \), so \( (1, -1) \); at \( x = 2 \), \( f(2) = 2 - 2 = 0 \), so \( (2, 0) \). Then draw a line segment from the open circle at \( (-1, -3) \) through these points to the closed circle at \( (3, 1) \).

(Note: Since this is a graphing problem, the final answer is the visual graph as described above. However, in text form, we can describe the key points and segments.)

Answer:

• For \( -4 \leq x \leq -1 \): Horizontal line segment between \( (-4, -2) \) (closed circle) and \( (-1, -2) \) (closed circle).
• For \( -1 < x \leq 3 \): Line segment with an open circle at \( (-1, -3) \) and a closed circle at \( (3, 1) \), passing through points like \( (0, -2) \), \( (1, -1) \), \( (2, 0) \).