Question

use the rule for function $f$ to draw its graph.
$f(x)=\

$$\begin{cases} 2, & -5\\leq x < -2 \\\\ 6, & -3\\leq x < 4 \\\\ x, & 4\\leq x < 8 \\end{cases}$$

$

Explanation:

Step1: Analyze the first piece

The first piece of the function is \( f(x) = 2 \) for \( -5\leq x < -2 \). This is a horizontal line segment. We plot points where \( x \) ranges from -5 (inclusive) to -2 (exclusive) and \( y = 2 \). So we can mark the point (-5, 2) with a closed dot (since \( x = -5 \) is included) and (-2, 2) with an open dot (since \( x=-2 \) is not included) and draw a horizontal line between them.

Step2: Analyze the second piece

The second piece is \( f(x)=6 \) for \( -2\leq x < 4 \). This is also a horizontal line. We plot points where \( x \) ranges from -2 (inclusive) to 4 (exclusive) and \( y = 6 \). So we mark (-2, 6) with a closed dot (since \( x = -2 \) is included) and (4, 6) with an open dot (since \( x = 4 \) is not included) and draw a horizontal line between them.

Step3: Analyze the third piece

The third piece is \( f(x)=x \) for \( 4\leq x < 8 \). This is a linear function with a slope of 1. We find two points: when \( x = 4 \), \( f(4)=4 \) (closed dot since \( x = 4 \) is included), and when \( x = 8 \), \( f(8)=8 \) (open dot since \( x = 8 \) is not included). Then we draw a line segment connecting (4, 4) and (8, 8).

To draw the graph:

1. For \( -5\leq x < -2 \), draw a horizontal line at \( y = 2 \), starting at (-5, 2) (closed circle) and ending at (-2, 2) (open circle).
2. For \( -2\leq x < 4 \), draw a horizontal line at \( y = 6 \), starting at (-2, 6) (closed circle) and ending at (4, 6) (open circle).
3. For \( 4\leq x < 8 \), draw a line with slope 1, starting at (4, 4) (closed circle) and ending at (8, 8) (open circle).

Answer:

The graph consists of three parts: a horizontal segment \( y = 2 \) from \( (-5, 2) \) (closed) to \( (-2, 2) \) (open), a horizontal segment \( y = 6 \) from \( (-2, 6) \) (closed) to \( (4, 6) \) (open), and a line segment \( y=x \) from \( (4, 4) \) (closed) to \( (8, 8) \) (open).