Sketch the graph of the function using a window that gives a complete graph.

Question

​f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column 5 2nd Column if x greater than or equals 0 2nd Row 1st Column negative 5 2nd Column if x less than 0 EndMatrix

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Part 1
Choose the correct graph below.
A.
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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 (0, negative 5) and moves to the left, passing through the point (negative 10, negative 5). The second branch is a horizontal ray that starts the open point (0, 5) and moves to the right, passing through the point (10, 5).
B.
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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 (0, negative 5) and moves to the left, passing through the point (negative 10, negative 5). The second branch is a horizontal ray that starts the closed point (0, 5) and moves to the right, passing through the point (10, 5).
C.
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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 (0, negative 5) and moves to the right, passing through the point (10, negative 5). The second branch is a horizontal ray that starts the open point (0, 5) and moves to the left, passing through the point (negative 10, 5).
D.
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Answer 1

To sketch and analyze the function described, we note that it’s a piecewise function defined as follows:

• $f(x)=5$ for $x\ge 0$
• $f(x)=-5$ for $x<0$

Breaking Down the Function

1. For $x\ge 0$:

   • The function takes the value $5$.
   • The graph includes the point (0, 5), which is a closed point (meaning this value is included).
   • The graph extends to the right (positive x-axis) indefinitely at $y=5$.

2. For $x<0$:

   • The function takes the value $-5$.
   • The graph does not include $(0,-5)$ which is an open point at $x=0$.
   • The graph extends to the left (negative x-axis) indefinitely at $y=-5$.

Sketching the Graph

• We will have:
  • A horizontal line at $y=5$ starting from $(0,5)$ and extending to the right.
  • A horizontal line at $y=-5$ starting from $(0,-5)$ extending to the left.

Analyzing Provided Options

Now let's analyze which option corresponds to our understanding:

1. Option A: The graph has a horizontal ray starting at (0, -5) to the left (which is incorrect, it should be at (0, 5)).
2. Option B: The graph has a horizontal ray starting at the open point (0, -5) to the left (which is incorrect, it should be at (0, 5)).
3. Option C: The graph has a horizontal ray starting at the closed point (0, -5) moving to the right (which is incorrect, it should be at (0, 5)).
4. Option D: The graph has a horizontal ray starting at the closed point (0, 5) moving to the right, which matches our findings and gives the correct representation of the function for $x\ge 0$ and a line at $y=-5$ extending to the left without including the point (0, -5).

Therefore, the correct option is:

D.