Question

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

$$\begin{cases} 3 & \\text{for} & -4 < x \\leq -1 \\\\ x + 3 & \\text{for} & -1 < x < 5 \\end{cases}$$

$

Explanation:

Step1: Analyze the first piece

The first piece of the function is \( f(x) = 3 \) for \( -4 < x \leq -1 \). This is a horizontal line segment. We need to plot the points:

• At \( x = -1 \), \( f(-1) = 3 \) (closed circle since \( x = -1 \) is included).
• For \( x \) values between \( -4 \) (open circle, since \( x = -4 \) is not included) and \( -1 \), the \( y \)-value is 3. So we draw a horizontal line from \( (-4, 3) \) (open circle) to \( (-1, 3) \) (closed circle).

Step2: Analyze the second piece

The second piece is \( f(x) = x + 3 \) for \( -1 < x < 5 \). This is a linear function. Let's find some key points:

• At \( x = -1 \), \( f(-1) = -1 + 3 = 2 \) (open circle since \( x = -1 \) is not included here).
• At \( x = 5 \), \( f(5) = 5 + 3 = 8 \) (open circle since \( x = 5 \) is not included).
• Let's also find the value at \( x = 0 \): \( f(0) = 0 + 3 = 3 \).
• At \( x = 2 \): \( f(2) = 2 + 3 = 5 \).
• We draw a line segment starting from \( (-1, 2) \) (open circle) with a slope of 1 (since the coefficient of \( x \) is 1) and ending at \( (5, 8) \) (open circle), passing through the points we calculated.

Graph Description:

• For \( -4 < x \leq -1 \): Horizontal line from \( (-4, 3) \) (open) to \( (-1, 3) \) (closed).
• For \( -1 < x < 5 \): Line with slope 1, starting at \( (-1, 2) \) (open) and ending at \( (5, 8) \) (open), passing through points like \( (0, 3) \), \( (2, 5) \) etc.

(Note: Since the problem asks to graph, the above steps describe how to construct the graph. If we were to represent the graph, we would have the two segments as described.)

Answer:

The graph consists of a horizontal segment from \( (-4, 3) \) (open circle) to \( (-1, 3) \) (closed circle) and a linear segment from \( (-1, 2) \) (open circle) to \( (5, 8) \) (open circle) with a slope of 1, passing through points like \( (0, 3) \), \( (2, 5) \) etc.