Answer :
Sure, let's go through the problem step by step.
### Step 1: Understand the function \( f(x) \)
The function \( f(x) \) is given as a piecewise function with different constant values in different intervals:
1. \( f(x) = -3 \) for \(-2 < x \leq -1\)
2. \( f(x) = -2 \) for \(-1 < x \leq 0\)
3. \( f(x) = -1 \) for \( 0 < x \leq 1 \)
### Step 2: Plotting the points
We'll plot each segment of the piecewise function.
- For the first segment \( f(x) = -3 \):
- This applies to the interval \(-2 < x \leq -1\).
- At \( x = -1 \), \( f(x) = -3 \), and there should be a closed circle at \((-1, -3)\).
- Since it is not defined at \( x = -2 \), there should be an open circle at \((-2, -3)\).
- For the second segment \( f(x) = -2 \):
- This applies to the interval \(-1 < x \leq 0\).
- At \( x = 0 \), \( f(x) = -2 \), and there should be a closed circle at \((0, -2)\).
- At \( x = -1 \), \( f(x) = -2 \), but there should be an open circle just to the right of \( x = -1 \).
- For the third segment \( f(x) = -1 \):
- This applies to the interval \( 0 < x \leq 1 \).
- At \( x = 1 \), \( f(x) = -1 \), and there should be a closed circle at \((1, -1)\).
- At \( x = 0 \), \( f(x) = -1 \), but there should be an open circle just to the right of \( x = 0 \).
### Step 3: Drawing the graph
1. First Interval \(-2 < x \leq -1\):
- Draw a horizontal line at \( y = -3 \) from just to the right of \( x = -2 \) to \( x = -1 \).
- Mark \((-2, -3)\) with an open circle.
- Mark \((-1, -3)\) with a closed circle.
2. Second Interval \(-1 < x \leq 0\):
- Draw a horizontal line at \( y = -2 \) from just to the right of \( x = -1 \) to \( x = 0 \).
- Mark \((-1, -2)\) with an open circle.
- Mark \((0, -2)\) with a closed circle.
3. Third Interval \( 0 < x \leq 1\):
- Draw a horizontal line at \( y = -1 \) from just to the right of \( x = 0 \) to \( x = 1 \).
- Mark \((0, -1)\) with an open circle.
- Mark \((1, -1)\) with a closed circle.
### Step 4: Verify the options
Compare the graph you have drawn with the descriptions provided in the answer choices. Each choice should describe the intervals, the positions of the open and closed circles accurately.
### Step 5: Determine the correct answer
To match the graph fully, look for the choice that aligns with the details:
- Closed circle at specific points: (\(-1, -3\), (0, -2), (1, -1))
- Open circle at specific points: (\((-2, -3)\), \((-1, -2)\), \((0, -1)\))
I cannot provide the specific correct choice without the descriptions, but you should now be able to match the graph characteristics with the correct option.
### Step 1: Understand the function \( f(x) \)
The function \( f(x) \) is given as a piecewise function with different constant values in different intervals:
1. \( f(x) = -3 \) for \(-2 < x \leq -1\)
2. \( f(x) = -2 \) for \(-1 < x \leq 0\)
3. \( f(x) = -1 \) for \( 0 < x \leq 1 \)
### Step 2: Plotting the points
We'll plot each segment of the piecewise function.
- For the first segment \( f(x) = -3 \):
- This applies to the interval \(-2 < x \leq -1\).
- At \( x = -1 \), \( f(x) = -3 \), and there should be a closed circle at \((-1, -3)\).
- Since it is not defined at \( x = -2 \), there should be an open circle at \((-2, -3)\).
- For the second segment \( f(x) = -2 \):
- This applies to the interval \(-1 < x \leq 0\).
- At \( x = 0 \), \( f(x) = -2 \), and there should be a closed circle at \((0, -2)\).
- At \( x = -1 \), \( f(x) = -2 \), but there should be an open circle just to the right of \( x = -1 \).
- For the third segment \( f(x) = -1 \):
- This applies to the interval \( 0 < x \leq 1 \).
- At \( x = 1 \), \( f(x) = -1 \), and there should be a closed circle at \((1, -1)\).
- At \( x = 0 \), \( f(x) = -1 \), but there should be an open circle just to the right of \( x = 0 \).
### Step 3: Drawing the graph
1. First Interval \(-2 < x \leq -1\):
- Draw a horizontal line at \( y = -3 \) from just to the right of \( x = -2 \) to \( x = -1 \).
- Mark \((-2, -3)\) with an open circle.
- Mark \((-1, -3)\) with a closed circle.
2. Second Interval \(-1 < x \leq 0\):
- Draw a horizontal line at \( y = -2 \) from just to the right of \( x = -1 \) to \( x = 0 \).
- Mark \((-1, -2)\) with an open circle.
- Mark \((0, -2)\) with a closed circle.
3. Third Interval \( 0 < x \leq 1\):
- Draw a horizontal line at \( y = -1 \) from just to the right of \( x = 0 \) to \( x = 1 \).
- Mark \((0, -1)\) with an open circle.
- Mark \((1, -1)\) with a closed circle.
### Step 4: Verify the options
Compare the graph you have drawn with the descriptions provided in the answer choices. Each choice should describe the intervals, the positions of the open and closed circles accurately.
### Step 5: Determine the correct answer
To match the graph fully, look for the choice that aligns with the details:
- Closed circle at specific points: (\(-1, -3\), (0, -2), (1, -1))
- Open circle at specific points: (\((-2, -3)\), \((-1, -2)\), \((0, -1)\))
I cannot provide the specific correct choice without the descriptions, but you should now be able to match the graph characteristics with the correct option.
