To graph the piecewise function
[tex]\[ r(x) = \begin{cases}
1 & \text{if } x < -1 \\
-4 & \text{if } x \geq -1
\end{cases} \][/tex]
we need to break it down into the two different parts specified by the piecewise definition:
1. For [tex]\( x < -1 \)[/tex]:
The value of [tex]\( r(x) \)[/tex] is 1.
This part of the function is a horizontal line at [tex]\( r(x) = 1 \)[/tex] for all values of [tex]\( x \)[/tex] that are strictly less than -1.
2. For [tex]\( x \geq -1 \)[/tex]:
The value of [tex]\( r(x) \)[/tex] is -4.
This part of the function is a horizontal line at [tex]\( r(x) = -4 \)[/tex] for all values of [tex]\( x \)[/tex] that are greater than or equal to -1.
Now let's graph this step by step:
### Step-by-Step Graphing:
1. Draw a horizontal line at [tex]\( r(x) = 1 \)[/tex] for values of [tex]\( x \)[/tex] less than -1. This line is continuous to the left side of [tex]\( x = -1 \)[/tex] but it does not include [tex]\( x = -1 \)[/tex]. To indicate that the point [tex]\((-1, 1)\)[/tex] is not included, we use an open circle at [tex]\( x = -1 \)[/tex].
2. Draw a horizontal line at [tex]\( r(x) = -4 \)[/tex] for values of [tex]\( x \)[/tex] greater than or equal to -1. We start this part of the graph at [tex]\((-1, -4)\)[/tex] and use a closed circle at [tex]\( x = -1 \)[/tex] to indicate that this point is included in the graph.
Combining these steps, the correct piecewise graph will show:
- An open circle at [tex]\( (-1, 1) \)[/tex] and a horizontal line extending to the left for [tex]\( x < -1 \)[/tex] at the height 1.
- A closed circle at [tex]\( (-1, -4) \)[/tex] and a horizontal line extending to the right for [tex]\( x \geq -1 \)[/tex] at the height -4.
Based on this explanation, you need a graph that matches these characteristics.
If shown with options A, B, C, and D:
- Select the graph that correctly represents these conditions with an open circle at [tex]\( (-1, 1) \)[/tex], a horizontal line for [tex]\( x < -1 \)[/tex] at [tex]\( y = 1 \)[/tex], a closed circle at [tex]\( (-1, -4) \)[/tex], and a horizontal line for [tex]\( x \geq -1 \)[/tex] at [tex]\( y = -4 \)[/tex].
