Question
Sketch the graph of the function using a window that gives a complete graph.
f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column negative 6 2nd Column if x greater than or equals minus 2 2nd Row 1st Column 6 2nd Column if x less than minus 2 EndMatrix
Question content area bottom
Part 1
Choose the correct graph below.
A.
-10
10
-10
10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. A graph has two branches. The first branch is a horizontal ray that startes at the open point (negative 2, 6) and moves to the right, passing through the point (10, 6). The second branch is a horizontal ray that starts the closed point (negative 2, negative 6) and moves to the left, passing through the point (negative 10, negative 6).
B.
-10
10
-10
10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. A graph has two branches. The first branch is a horizontal ray that startes at the closed point (negative 2, 6) and moves to the right, passing through the point (10, 6). The second branch is a horizontal ray that starts the open point (negative 2, negative 6) and moves to the left, passing through the point (negative 10, negative 6).
C.
-10
10
-10
10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. A graph has two branches. The first branch is a horizontal ray that startes at the closed point (negative 2, 6) and moves to the left, passing through the point (negative 10, 6). The second branch is a horizontal ray that starts the open point (negative 2, negative 6) and moves to the right, passing through the point (10, negative 6).
D.
-10
10
-10
10
f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column negative 6 2nd Column if x greater than or equals minus 2 2nd Row 1st Column 6 2nd Column if x less than minus 2 EndMatrix
Question content area bottom
Part 1
Choose the correct graph below.
A.
-10
10
-10
10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. A graph has two branches. The first branch is a horizontal ray that startes at the open point (negative 2, 6) and moves to the right, passing through the point (10, 6). The second branch is a horizontal ray that starts the closed point (negative 2, negative 6) and moves to the left, passing through the point (negative 10, negative 6).
B.
-10
10
-10
10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. A graph has two branches. The first branch is a horizontal ray that startes at the closed point (negative 2, 6) and moves to the right, passing through the point (10, 6). The second branch is a horizontal ray that starts the open point (negative 2, negative 6) and moves to the left, passing through the point (negative 10, negative 6).
C.
-10
10
-10
10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. A graph has two branches. The first branch is a horizontal ray that startes at the closed point (negative 2, 6) and moves to the left, passing through the point (negative 10, 6). The second branch is a horizontal ray that starts the open point (negative 2, negative 6) and moves to the right, passing through the point (10, negative 6).
D.
-10
10
-10
10
Answers
GPT-4o mini
To analyze the piecewise function given:
\[
f(x) =
\begin{cases}
-6 & \text{if } x \geq -2 \\
6 & \text{if } x < -2
\end{cases}
\]
1. For \(x \geq -2\), \(f(x) = -6\). This means that there is a horizontal line at \(y = -6\) starting from \(x = -2\) and extending to the right. Since the inequality is inclusive, the point \((-2, -6)\) will be a closed point.
2. For \(x < -2\), \(f(x) = 6\). This means there is another horizontal line at \(y = 6\) starting from \(x = -2\) and extending to the left. Since this part is exclusive, the point \((-2, 6)\) will be an open point.
Now let's examine each option:
- **Option A:**
- Closed point at \((-2, -6)\) moving right → Incorrect.
- Open point at \((-2, 6)\) moving left → Incorrect.
- **Option B:**
- Closed point at \((-2, 6)\) moving right → Correct.
- Open point at \((-2, -6)\) moving left → Incorrect.
- **Option C:**
- Closed point at \((-2, 6)\) moving left → Incorrect.
- Open point at \((-2, -6)\) moving right → Correct.
- **Option D:**
- Closed point at \((-2, 6)\) moving left → Incorrect.
- Open point at \((-2, -6)\) moving right → Correct.
None of the options perfectly match the description. However, the closest correct representation is **Option B**, which accurately shows the horizontal line at \(y = -6\) starting from \((-2, -6)\) moving to the right and the horizontal line at \(y = 6\) starting from \((-2, 6)\) moving to the left.
Therefore, the correct choice is **Option B**.
\[
f(x) =
\begin{cases}
-6 & \text{if } x \geq -2 \\
6 & \text{if } x < -2
\end{cases}
\]
1. For \(x \geq -2\), \(f(x) = -6\). This means that there is a horizontal line at \(y = -6\) starting from \(x = -2\) and extending to the right. Since the inequality is inclusive, the point \((-2, -6)\) will be a closed point.
2. For \(x < -2\), \(f(x) = 6\). This means there is another horizontal line at \(y = 6\) starting from \(x = -2\) and extending to the left. Since this part is exclusive, the point \((-2, 6)\) will be an open point.
Now let's examine each option:
- **Option A:**
- Closed point at \((-2, -6)\) moving right → Incorrect.
- Open point at \((-2, 6)\) moving left → Incorrect.
- **Option B:**
- Closed point at \((-2, 6)\) moving right → Correct.
- Open point at \((-2, -6)\) moving left → Incorrect.
- **Option C:**
- Closed point at \((-2, 6)\) moving left → Incorrect.
- Open point at \((-2, -6)\) moving right → Correct.
- **Option D:**
- Closed point at \((-2, 6)\) moving left → Incorrect.
- Open point at \((-2, -6)\) moving right → Correct.
None of the options perfectly match the description. However, the closest correct representation is **Option B**, which accurately shows the horizontal line at \(y = -6\) starting from \((-2, -6)\) moving to the right and the horizontal line at \(y = 6\) starting from \((-2, 6)\) moving to the left.
Therefore, the correct choice is **Option B**.